#6575578

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#36009	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2207 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482774);