#6668931

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	#23871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2058 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)(7,10)(8,9)(11,12)(16,27)(20,37)(25,29)(28,40)(30,33)(36,38)(41,44)(42,43);(6,9)(7,11)(13,14,18,22)(15,23,24,39)(16,28,27,40)(17,31,34,21)(19,32,35,26)(20,33,37,30)(25,36,29,38)(41,43,44,42);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,29)(17,32)(18,24)(20,38)(21,39)(22,35)(23,31)(25,27)(26,34)(28,44)(30,42)(33,43)(36,37)(40,41);(6,9)(10,12)(15,24)(19,35)(23,39)(25,29)(26,32)(36,38)(41,44)(42,43);(13,16,18,27)(14,20,22,37)(15,25,24,29)(17,33,34,30)(19,36,35,38)(21,40,31,28)(23,41,39,44)(26,43,32,42)

Polycyclic presentation:

2^9|b^2=i,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6576127);