#6574784

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2024 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);(6,9)(10,12)(13,14,18,23)(15,20,26,35)(16,27,30,40)(17,19,31,22)(21,33,38,29)(24,39,32,34)(25,37,42,43)(28,36,41,44);(2,4)(5,7,8,11)(6,10,9,12)(14,19)(15,24)(17,31)(20,35)(21,27)(22,23)(25,41)(26,32)(28,42)(29,33)(36,44)(38,40);(13,15,18,26)(14,20,23,35)(16,28,30,41)(17,32,31,24)(19,34,22,39)(21,37,38,43)(25,33,42,29)(27,36,40,44);(13,16,18,30)(14,21,23,38)(15,25,26,42)(17,33,31,29)(19,27,22,40)(20,36,35,44)(24,41,32,28)(34,43,39,37)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,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, 6481980);