#7615642

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1343 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(19,28)(20,32)(24,29)(25,41)(31,36)(33,46)(34,35)(39,40);(2,6)(7,8)(11,16)(13,15)(17,18,22,27)(19,29,28,24)(20,33,32,46)(21,26,23,37)(25,34,41,35)(30,44,42,47)(31,40,36,39)(38,45,43,48);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19,22,28)(18,24,27,29)(20,30,32,42)(21,36,23,31)(25,38,41,43)(26,40,37,39)(33,47,46,44)(34,48,35,45);(10,14)(13,15)(17,20,21,34,22,32,23,35)(18,25,37,46,27,41,26,33)(19,30,36,48,28,42,31,45)(24,38,39,44,29,43,40,47);(1,4,5,3)(2,7,6,8)(17,21,22,23)(18,26,27,37)(19,31,28,36)(20,34,32,35)(24,39,29,40)(25,33,41,46)(30,45,42,48)(38,44,43,47)

Polycyclic presentation:

2^9|a^2=gi,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,e^c=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, 7522838);