#6617457

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2995 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)(13,14,17,22,19,24,18,23)(15,25,27,20,29,35,28,36)(16,30,33,21,32,37,31,38)(26,34,41,39,42,44,40,43);(2,4)(6,10)(8,11)(13,15)(14,20)(16,26)(17,28)(18,27)(19,29)(21,34)(22,25)(23,35)(24,36)(30,39)(31,41)(32,42)(33,40)(37,43)(38,44);(5,6)(7,8)(9,10)(11,12)(13,16,19,32)(14,21,24,38)(15,26,29,42)(17,31,18,33)(20,34,36,44)(22,30,23,37)(25,39,35,43)(27,40,28,41);(5,7,9,12)(6,11,10,8)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,31,32,33)(20,35,36,25)(21,30,38,37)(26,40,42,41)(34,43,44,39);(5,8,9,11)(6,7,10,12)(13,17,19,18)(14,23,24,22)(15,28,29,27)(16,31,32,33)(20,35,36,25)(21,37,38,30)(26,41,42,40)(34,43,44,39)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=hi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,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, 6524653);