#6617847

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#39131	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1811 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6525043);