#6576839

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	#35757	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)(7,12)(8,10)(9,11)(13,14,18,22)(15,19,25,33)(16,20,28,36)(17,21,29,37)(23,31,30,38)(24,32,40,43)(26,34,27,35)(39,42,41,44);(5,7,9,8)(6,10,11,12)(15,23)(16,26)(17,29)(19,31)(20,34)(21,37)(25,30)(27,28)(33,38)(35,36)(39,41)(42,44);(2,4)(13,15,18,25)(14,19,22,33)(16,24,28,40)(17,30,29,23)(20,32,36,43)(21,38,37,31)(26,39,27,41)(34,42,35,44);(5,8,9,7)(6,10,11,12)(15,23)(16,27)(17,29)(19,31)(20,35)(21,37)(24,40)(25,30)(26,28)(32,43)(33,38)(34,36);(13,16,18,28)(14,20,22,36)(15,24,25,40)(17,27,29,26)(19,32,33,43)(21,35,37,34)(23,39,30,41)(31,42,38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=ehi,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, 6484035);