#72339945

order	4096
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁷
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 19184 conjugacy classes of subgroups; only the 65379 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 28 points:

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

Polycyclic presentation:

2^12|b^2=h,c^2=l,b^a=bh,d^c=dl,g^a=gi,g^b=gh,g^c=gk,g^d=gj,g^e=gijkl,g^f=ghl

Conjugacy classes & character table

520 conjugacy classes — view the full character table.

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", "k", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.12);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 7, 6, F.7*F.8*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);