#72326355

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 10363 conjugacy classes of subgroups; only the 3329 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:

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

Polycyclic presentation:

2^10|b^2=i,c^2=j,e^2=ghij,b^a=bg,c^b=cgh,d^b=di,d^c=di,e^b=eghj,e^c=eghi,e^d=ei,f^a=fj,f^b=fh,f^c=fj,f^d=fh

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326378, #72329191
maximal normal subgroup of	#72326378, #72329191
maximal quotient of	#72333338

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, 2, F.9);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 5, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);