#6631590

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1214 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)(9,10,13,14)(11,15,18,21)(12,16,20,22)(17,23,19,24)(25,26)(27,32)(28,40)(29,34)(30,43)(31,35)(33,41)(36,50)(37,46)(38,47)(39,53)(42,48)(44,49)(45,52)(51,56)(54,55);(1,3,4,8)(2,5,6,7)(10,14)(15,21)(16,22)(23,24)(25,27)(26,32)(28,36)(29,38)(30,39)(31,37)(33,45)(34,47)(35,46)(40,50)(41,52)(42,51)(43,53)(44,54)(48,56)(49,55);(1,2)(3,7)(4,6)(5,8)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,28)(26,33)(27,36)(29,41)(30,44)(31,42)(32,45)(34,48)(35,49)(37,51)(38,52)(39,54)(40,43)(46,55)(47,56)(50,53);(2,6)(5,7)(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(27,37)(32,46)(36,51)(38,39)(45,55)(47,53)(50,56)(52,54);(25,29,31,30)(26,34,35,43)(27,38,37,39)(28,41,42,44)(32,47,46,53)(33,48,49,40)(36,52,51,54)(45,56,55,50)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,d^c=dfh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538786);