#512179

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

643 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=e,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419375);