#501801

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	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

698 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=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, 408997);