#6775678

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#31389	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1470 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)(19,29)(20,34)(21,27)(22,28)(23,25)(24,32)(26,37)(30,40)(31,39)(33,41)(35,43)(36,44)(38,42)(45,47)(46,48);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,32)(18,24,28,29)(20,35,37,30)(21,31,23,33)(25,39,27,41)(26,43,34,40)(36,46,38,45)(42,48,44,47);(10,13)(14,16)(18,25)(21,23)(24,39)(26,42)(27,28)(29,41)(31,33)(34,44)(36,38)(40,47)(43,48)(45,46);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20,22,37)(18,26,28,34)(19,30,32,35)(21,38,23,36)(24,40,29,43)(25,42,27,44)(31,46,33,45)(39,47,41,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21,22,23)(18,27,28,25)(19,31,32,33)(20,36,37,38)(24,41,29,39)(26,42,34,44)(30,45,35,46)(40,47,43,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6682874);