#72267006

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

167396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^11|c^2=ghj,d^2=hij,e^2=k,f^2=k,c^a=cg,c^b=cgh,d^a=di,d^b=dh,d^c=dj,e^c=ejk,f^d=fjk,f^e=fk

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72267018, #72267024
Fitting subgroup of	#72267018
maximal subgroup of	#72267008, #72267018
maximal normal subgroup of	#72267008, #72267018

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