#6773488

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,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,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6680684);