#6640446

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

1248 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)(27,32)(28,39)(29,34)(30,44)(31,35)(33,41)(36,45)(37,51)(38,50)(40,53)(42,54)(43,47)(46,52)(48,55)(49,56);(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)(26,32)(28,40)(29,31)(30,38)(33,48)(34,35)(36,47)(37,42)(39,53)(41,55)(43,45)(44,50)(46,49)(51,54)(52,56);(3,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28,30,41,31,42,45,56)(26,33,36,39,35,49,44,54)(27,37,38,52,29,40,43,55)(32,46,47,51,34,48,50,53);(3,7)(5,8)(10,17)(11,20)(16,22)(19,21)(25,29,31,27)(26,34,35,32)(28,37,42,40)(30,43,45,38)(33,46,49,48)(36,50,44,47)(39,51,54,53)(41,52,56,55);(2,6)(5,8)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(26,35)(27,29)(33,49)(36,44)(37,40)(38,43)(39,54)(52,55)

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=i,e^2=f,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^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, 6547642);