#6605533

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#37659	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1969 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,28,31,39)(20,32,33,25)(21,26,22,36)(24,37,40,30)(29,43,44,42)(34,47,35,41)(38,45,46,48);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,29)(21,30)(22,37)(23,31)(25,38)(26,39)(27,40)(28,36)(32,46)(33,44)(34,45)(35,48)(41,43)(42,47);(2,6)(5,8)(10,14)(11,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,38)(26,41)(27,32)(28,42)(30,45)(31,44)(36,47)(37,48)(39,43)(40,46);(9,11)(10,13)(12,15)(14,16)(20,33)(25,32)(29,44)(34,35)(38,46)(41,47)(42,43)(45,48);(11,15)(13,16)(17,21,23,22)(18,26,27,36)(19,30,31,37)(20,34,33,35)(24,39,40,28)(25,41,32,47)(29,45,44,48)(38,43,46,42)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6512729);