#6673220

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#24390	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2083 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6580416);