#6611996

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1691 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)(9,10)(11,14)(12,16)(13,15)(17,18,23,35,22,26,21,27)(19,28,34,24,31,41,30,38)(20,25,36,46,32,39,33,40)(29,42,45,37,43,48,44,47);(9,11,13,12)(10,14,15,16)(17,19,22,31)(18,24,26,38)(20,29,32,43)(21,34,23,30)(25,37,39,47)(27,41,35,28)(33,45,36,44)(40,48,46,42);(2,5)(6,8)(10,15)(11,12)(17,20)(18,25)(19,29)(21,33)(22,32)(23,36)(24,37)(26,39)(27,40)(28,42)(30,44)(31,43)(34,45)(35,46)(38,47)(41,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,15,14)(20,32)(25,39)(29,43)(33,36)(37,47)(40,46)(42,48)(44,45);(11,12)(14,16)(18,26)(24,38)(25,39)(27,35)(28,41)(37,47)(40,46)(42,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6519192);