#501359

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	C₂×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

758 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)(7,10)(8,9)(11,12)(13,14)(15,24)(16,29)(17,21)(18,23)(19,22)(20,27)(25,34)(26,35)(28,42)(30,41)(31,38)(32,39)(33,37)(36,43)(40,44);(6,9)(10,12)(13,15,17,27)(14,20,21,24)(16,26,19,33)(18,25,31,44)(22,35,29,37)(23,40,38,34)(28,30,43,32)(36,41,42,39);(2,4)(5,7,8,11)(6,10,9,12)(14,21)(15,25)(16,19)(18,32)(20,34)(23,41)(24,40)(26,28)(27,44)(30,31)(33,43)(35,36)(37,42)(38,39);(6,9)(10,12)(13,16,17,19)(14,22,21,29)(15,26,27,33)(18,30,31,32)(20,35,24,37)(23,39,38,41)(25,43,44,28)(34,42,40,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=ei,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, 408555);