#511980

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

1008 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)(5,6,8,10)(7,11,12,9)(13,14,17,20)(15,24,26,41)(16,22,19,33)(18,30,29,43)(21,37,35,27)(23,28,36,32)(25,42,34,40)(31,38,44,39);(5,7,8,12)(6,9,10,11)(14,20)(21,35)(22,33)(23,36)(24,41)(30,43)(38,39)(40,42);(13,15)(14,21)(16,25)(17,26)(18,31)(19,34)(20,35)(22,38)(23,40)(24,30)(27,32)(28,37)(29,44)(33,39)(36,42)(41,43);(2,4)(13,16,17,19)(14,22,20,33)(15,25,26,34)(18,28,29,32)(21,38,35,39)(23,30,36,43)(24,42,41,40)(27,31,37,44)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,f^c=fh,h^c=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, 419176);