#72331856

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 9967 conjugacy classes of subgroups; only the 3317 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:

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

Polycyclic presentation:

2^10|d^2=g,b^a=bi,c^b=ch,d^a=dij,d^b=dg,d^c=dij,e^a=eij,e^c=eij,e^d=eg,f^a=fij,f^b=fhij,f^d=fghi,f^e=fhij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331930, #72333429
maximal normal subgroup of	#72331930, #72333429
maximal quotient of	#72333260

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