#7410316

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1283 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)(3,6)(4,7)(5,8)(9,10,13,12)(11,14,15,16)(17,18)(19,29)(20,26)(21,28)(22,24)(23,27)(25,33)(30,41)(31,45)(32,38)(34,48)(35,44)(36,39)(37,43)(40,46)(42,47);(1,3,6,2)(4,7,5,8)(9,11)(10,14)(12,16)(13,15)(18,24)(19,30)(26,39)(27,28)(31,34)(32,33)(43,44)(46,47);(10,12)(11,15)(17,19,22,30)(18,25,24,38)(20,34,36,31)(21,33,23,32)(26,42,39,40)(27,29,28,41)(35,46,37,47)(43,45,44,48);(1,4,6,5)(2,7,3,8)(9,12,13,10)(11,14,15,16)(17,20)(18,26)(19,31)(21,37)(22,36)(23,35)(24,39)(25,40)(27,44)(28,43)(29,45)(30,34)(32,47)(33,46)(38,42)(41,48);(1,5,6,4)(2,7,3,8)(11,15)(14,16)(17,21,22,23)(18,27,24,28)(19,32,30,33)(20,35,36,37)(25,41,38,29)(26,43,39,44)(31,46,34,47)(40,45,42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=i,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,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, 7317512);