#512085

order	512
nilpotency class	4
isoclinism	isoclinic class of #511741
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1026 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,h^a=hi,f^c=fh,g^c=gh,h^c=hi,f^d=fh,g^d=gh,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, 419281);