#6534058

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,d^c=di,e^c=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6441254);