#504374

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

556 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)(7,10)(8,9)(11,12)(13,14,20,26)(15,27,16,22)(17,25,19,37)(18,33,36,43)(21,31,40,29)(23,35,24,32)(28,38,30,42)(34,41,44,39);(1,2)(3,4)(13,15,20,16)(14,21,26,40)(17,30,19,28)(18,34,36,44)(22,33,27,43)(23,42,24,38)(25,41,37,39)(29,32,31,35);(2,4)(6,9)(10,12)(13,16,20,15)(14,22,26,27)(17,30,19,28)(18,31,36,29)(21,33,40,43)(23,41,24,39)(25,42,37,38)(32,34,35,44);(5,7,8,11)(6,10,9,12)(13,17)(14,23)(15,28)(16,30)(19,20)(21,38)(22,41)(24,26)(25,33)(27,39)(32,35)(34,44)(37,43)(40,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411570);