#72329059

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 10243 conjugacy classes of subgroups; only the 3309 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(2,3)(5,6)(7,8)(9,10)(11,12)(22,25)(26,27);(5,7)(6,8)(9,11)(10,12)(21,22)(23,27)(24,25)(26,28);(1,2,4,3)(6,10)(7,11)(14,17)(18,19)(21,23,24,28)(22,26,25,27);(1,2)(3,4)(6,10)(8,12)(13,14,16,17)(15,18,20,19)(21,23,24,28)(22,27,25,26);(2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,19)(16,17)(18,20)(21,23)(22,27)(24,28)(25,26);(1,3,4,2)(5,8)(6,7)(9,12)(10,11)(13,15,16,20)(14,18,17,19)(21,22)(23,27)(24,25)(26,28)

Polycyclic presentation:

2^10|c^2=hi,d^2=ghj,f^2=gij,b^a=bg,c^a=cgi,c^b=cgh,d^a=dgi,d^c=dgij,e^c=ehij,e^d=ehij,f^a=fhi,f^c=fg,f^d=fgi,f^e=fgij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329191, #72332061
maximal normal subgroup of	#72329191, #72332061
maximal quotient of	#72333258

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.8*F.9);;
gap> SetPower(coll, 4, F.7*F.8*F.10);;
gap> SetPower(coll, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);