#7620670

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	C₂²×D₄	C₂⁴
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) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fghi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,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, 7527866);