#7605226

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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

1388 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=fh,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,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, 7512422);