#6501990

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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

1670 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)(17,18,21,26)(19,28,30,23)(20,25,32,39)(22,24,34,27)(29,41,42,36)(31,37,44,35)(33,38,45,40)(43,47,48,46);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,30)(18,23,26,28)(20,29,32,42)(22,31,34,44)(24,35,27,37)(25,36,39,41)(33,43,45,48)(38,46,40,47);(9,11)(10,13)(12,16)(14,15)(18,24)(22,34)(23,35)(25,38)(26,27)(28,37)(31,44)(33,45)(36,46)(39,40)(41,47)(43,48);(11,16)(13,15)(17,20)(18,25)(19,29)(21,32)(22,33)(23,36)(24,38)(26,39)(27,40)(28,41)(30,42)(31,43)(34,45)(35,46)(37,47)(44,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(20,32)(25,39)(29,42)(33,45)(36,41)(38,40)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,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, 6409186);