#504442

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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

611 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)(13,14,17,23,33,41,43,39,19,25,34,38,18,24,32,22)(15,26,27,42,30,37,44,36,16,21,29,35,28,40,31,20);(1,2)(3,4)(13,15,19,16)(14,20,25,36)(17,31,34,44)(18,30,33,28)(21,22,26,39)(23,40,38,37)(24,42,41,35)(27,43,29,32);(2,4)(6,9)(10,12)(13,16,19,15)(14,21,25,26)(17,29,34,27)(18,30,33,28)(20,22,36,39)(23,35,38,42)(24,37,41,40)(31,32,44,43);(5,7,8,11)(6,10,9,12)(14,22)(17,32)(18,33)(20,26)(21,36)(23,24)(25,39)(27,31)(28,30)(29,44)(34,43)(35,37)(38,41)(40,42)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411638);