#7511869

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
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×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1600 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,26)(20,25)(21,27)(22,28)(23,29)(24,32)(30,41)(31,44)(33,48)(34,42)(35,43)(36,39)(37,45)(38,47)(40,46);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,32)(18,24,28,26)(20,33,35,46)(21,31,23,36)(25,40,43,48)(27,39,29,44)(30,37,47,34)(38,45,41,42);(10,14)(13,15)(17,20,22,35)(18,25,28,43)(19,30,32,47)(21,34,23,37)(24,38,26,41)(27,42,29,45)(31,46,36,33)(39,48,44,40);(2,7)(3,8)(11,16)(13,15)(17,19)(18,26)(20,30)(21,36)(22,32)(23,31)(24,28)(25,41)(27,39)(29,44)(33,34)(35,47)(37,46)(38,43)(40,45)(42,48);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,27,28,29)(19,31,32,36)(20,34,35,37)(24,39,26,44)(25,42,43,45)(30,46,47,33)(38,48,41,40)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,h^c=hi,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, 7419065);