#509668

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

604 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:

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

Polycyclic presentation:

2^9|a^2=ei,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,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, 416864);