#6575209

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	#35737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2179 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)(14,19)(20,35)(21,36)(22,33)(23,37)(28,40)(38,39)(41,44);(6,9)(10,12)(13,14,17,19)(15,23,26,37)(16,28,31,40)(18,33,32,22)(20,24,35,27)(21,34,36,30)(25,38,29,39)(41,43,44,42);(2,4)(13,15)(14,20)(16,29)(17,26)(18,24)(19,35)(21,39)(22,37)(23,33)(25,31)(27,32)(28,44)(30,42)(34,43)(36,38)(40,41);(5,7,8,11)(6,10,9,12)(15,24)(16,30)(18,32)(20,37)(21,28)(22,33)(23,35)(25,29)(26,27)(31,34)(36,40)(38,39);(13,16,17,31)(14,21,19,36)(15,25,26,29)(18,34,32,30)(20,38,35,39)(22,40,33,28)(23,41,37,44)(24,42,27,43)

Polycyclic presentation:

2^9|b^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482405);