#6640537

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636940
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1250 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,20,15)(12,16,13,22)(19,24,21,23)(25,26)(27,35)(28,39)(29,33)(30,44)(31,37)(32,42)(34,43)(36,50)(38,46)(40,53)(41,54)(45,55)(47,51)(48,56)(49,52);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24)(25,27,29,37)(26,31,33,35)(28,36,41,40)(30,38,43,52)(32,45,48,47)(34,46,44,49)(39,53,54,50)(42,51,56,55);(3,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28,30,42,29,41,43,56)(26,32,34,39,33,48,44,54)(27,36,38,51,37,40,52,55)(31,45,46,53,35,47,49,50);(3,7)(5,8)(10,17)(11,20)(16,22)(19,21)(25,27)(26,31)(28,40)(29,37)(30,38)(32,47)(33,35)(34,46)(36,41)(39,50)(42,55)(43,52)(44,49)(45,48)(51,56)(53,54);(2,6)(5,8)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(26,33)(27,37)(28,41)(34,44)(38,52)(42,56)(45,47)(50,53)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6547733);