#6954137

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1373 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2)(3,7)(4,6)(5,8)(10,14)(12,19)(15,21)(17,20)(25,26,30,35)(27,31,39,48)(28,33,40,51)(29,42,44,34)(32,43,49,41)(36,50,37,46)(38,54,55,47)(45,53,56,52);(1,3,4,8)(2,5,6,7)(9,10,13,14)(11,15,18,21)(12,16,19,22)(17,23,20,24)(25,27)(26,31)(28,36)(29,38)(30,39)(32,45)(33,50)(34,47)(35,48)(37,40)(41,52)(42,54)(43,53)(44,55)(46,51)(49,56);(3,8)(5,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28)(26,32)(27,36)(29,43)(30,40)(31,45)(33,42)(34,51)(35,49)(37,39)(38,53)(41,44)(46,47)(48,56)(50,54)(52,55);(2,6)(5,7)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(25,28)(26,33)(27,37)(29,41)(30,40)(31,46)(32,42)(34,49)(35,51)(36,39)(38,53)(43,44)(45,47)(48,50)(52,55)(54,56);(10,14)(15,21)(16,22)(23,24)(28,40)(32,49)(33,51)(36,37)(41,43)(45,56)(46,50)(52,53)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fh,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6861333);