#6444889

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2026 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,h^d=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, 6352085);