#7410254

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=i,d^2=fhi,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,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, 7317450);