#7594805

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#45136	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1578 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,26)(20,34)(21,36)(22,28)(23,30)(24,29)(25,32)(27,38)(31,43)(33,41)(35,40)(37,45)(39,44)(42,47)(46,48);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,32)(18,25,28,26)(20,24,33,23)(21,35,37,47)(27,43,44,48)(29,34,30,41)(31,38,46,39)(36,42,45,40);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,23,19,22,33,24,32)(18,26,29,34,28,25,30,41)(21,35,38,31,37,47,39,46)(27,42,36,43,44,40,45,48);(9,11,12,16)(10,14,13,15)(17,21,23,38,22,37,24,39)(18,27,29,36,28,44,30,45)(19,31,33,47,32,46,20,35)(25,40,41,48,26,42,34,43);(2,5)(6,8)(18,28)(21,37)(25,26)(29,30)(31,46)(34,41)(35,47)(38,39)

Polycyclic presentation:

2^9|b^2=i,c^2=fh,d^2=g,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,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7502001);