#504354

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	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

587 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)(13,14,18,23)(15,20,26,36)(16,19,29,33)(17,30,28,22)(21,37,32,27)(24,31,40,43)(25,41,39,35)(34,44,42,38);(1,2)(3,4)(14,19)(16,27)(17,28)(20,31)(21,22)(23,33)(24,38)(25,39)(29,37)(30,32)(34,35)(36,43)(40,44)(41,42);(2,4)(6,9)(10,12)(13,15,18,26)(14,20,23,36)(16,24,29,40)(17,25,28,39)(19,31,33,43)(21,34,32,42)(22,35,30,41)(27,38,37,44);(5,7,8,11)(6,10,9,12)(14,19)(16,27)(17,28)(20,31)(21,22)(23,33)(24,38)(25,39)(29,37)(30,32)(34,35)(36,43)(40,44)(41,42)

Polycyclic presentation:

2^9|a^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,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, 411550);