#7605218

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

1381 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)(17,18,21,23)(19,27,31,40)(20,32,35,43)(22,36,29,26)(24,39,37,28)(25,42,38,33)(30,44,34,41)(45,48,47,46);(2,6)(7,8)(9,10)(11,16)(12,15)(13,14)(18,23)(19,28)(20,33)(22,29)(24,40)(25,43)(27,37)(30,34)(31,39)(32,38)(35,42)(45,47);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,19,29,39,21,31,22,28)(18,24,36,40,23,37,26,27)(20,34,42,46,35,30,33,48)(25,44,43,45,38,41,32,47);(9,11,13,12)(10,16,14,15)(17,20,22,33,21,35,29,42)(18,25,26,32,23,38,36,43)(19,30,28,46,31,34,39,48)(24,41,27,45,37,44,40,47);(1,4,5,3)(2,7,6,8)(9,12,13,11)(10,15,14,16)(18,23)(19,31)(25,38)(26,36)(28,39)(30,34)(32,43)(46,48)

Polycyclic presentation:

2^9|a^2=fghi,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, 7512414);