#6578207

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2029 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)(5,6,9,10)(7,11,8,12)(16,27)(20,28)(24,41)(29,43)(30,31)(35,44)(38,42)(39,40);(6,10)(11,12)(13,14,18,22)(15,23,26,36)(16,28,27,20)(17,32,34,21)(19,33,37,25)(24,42,41,38)(29,44,43,35)(30,40,31,39);(2,4)(13,15,18,26)(14,19,22,37)(16,29,27,43)(17,33,34,25)(20,38,28,42)(21,36,32,23)(24,31,41,30)(35,39,44,40);(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,24)(17,30)(18,27)(19,35)(21,40)(22,28)(23,38)(25,29)(26,41)(31,34)(32,39)(33,43)(36,42)(37,44);(5,8,9,7)(6,12,10,11)(14,21)(16,30)(17,34)(19,36)(20,28)(22,32)(23,37)(24,29)(25,33)(27,31)(38,42)(41,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

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, 6485403);