#509312

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

612 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)(7,8)(9,11)(10,12)(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);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(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)(5,8,9,10)(6,7,11,12)(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);(6,11)(7,12)(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)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=e,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, 416508);