#7615585

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

1309 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13);(2,6)(7,8)(11,16)(13,15)(17,18,23,27)(19,24,30,41)(20,31,34,43)(21,26,22,37)(25,36,44,33)(28,45,32,48)(29,39,35,40)(38,46,42,47);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,32)(21,35)(22,29)(23,30)(25,42)(26,40)(27,41)(28,34)(31,48)(33,46)(36,47)(37,39)(38,44)(43,45);(10,14)(13,15)(17,20,21,36,23,34,22,33)(18,25,37,43,27,44,26,31)(19,28,35,46,30,32,29,47)(24,38,39,48,41,42,40,45);(1,4,5,3)(2,7,6,8)(17,21,23,22)(18,26,27,37)(19,29,30,35)(20,33,34,36)(24,39,41,40)(25,43,44,31)(28,46,32,47)(38,45,42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fg,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7522781);