#6667587

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#35687	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2696 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)(7,11)(8,10)(9,12);(5,7)(6,9)(8,12)(10,11)(13,14,18,22)(15,23,24,39)(16,27,30,40)(17,31,34,21)(19,32,35,26)(20,33,38,29)(25,36,28,37)(41,43,44,42);(2,4)(13,15)(14,19)(16,28)(17,32)(18,24)(20,37)(21,39)(22,35)(23,31)(25,30)(26,34)(27,44)(29,42)(33,43)(36,38)(40,41);(6,10)(9,11)(15,24)(19,35)(23,39)(25,28)(26,32)(36,37)(41,44)(42,43);(7,12)(9,11)(13,16,18,30)(14,20,22,38)(15,25,24,28)(17,33,34,29)(19,36,35,37)(21,40,31,27)(23,41,39,44)(26,43,32,42)

Polycyclic presentation:

2^9|b^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6574783);