#6640433

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636852
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1227 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)(30,43)(31,37)(32,41)(34,42)(36,49)(38,45)(40,54)(44,56)(46,55)(47,51)(48,52)(50,53);(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)(26,31)(28,36)(29,37)(30,38)(32,44)(33,35)(34,45)(39,53)(40,50)(41,51)(42,52)(43,48)(46,47)(49,54)(55,56);(3,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28,30,41,29,40,42,55)(26,32,34,39,33,46,43,54)(27,36,38,51,37,50,52,56)(31,44,45,53,35,47,48,49);(3,7)(5,8)(10,17)(11,20)(16,22)(19,21)(26,33)(27,37)(28,40)(34,43)(38,52)(41,55)(44,47)(49,53);(2,6)(5,8)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(26,33)(27,37)(32,46)(34,43)(36,50)(38,52)(39,54)(51,56)

Polycyclic presentation:

2^9|a^2=fi,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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6547629);