#7092652

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1279 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=ghi,d^2=f,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=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, 6999848);