#501289

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	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

719 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)(7,11)(8,10)(9,12)(13,14)(15,23)(16,19)(17,21)(18,22)(20,25)(24,34)(26,40)(27,32)(28,33)(29,39)(30,44)(31,36)(35,41)(37,43)(38,42);(5,7,8,12)(6,9,10,11)(14,19)(15,24)(17,29)(18,28)(20,34)(21,37)(23,35)(25,41)(26,42)(27,43)(30,31)(32,39)(36,38)(40,44);(2,4)(13,15)(14,20)(16,25)(17,30)(18,31)(19,23)(21,38)(22,40)(24,29)(26,28)(27,42)(32,44)(33,36)(34,37)(35,39)(41,43);(6,10)(9,11)(14,19)(15,25)(21,32)(22,33)(24,41)(26,31)(30,42)(37,39)

Polycyclic presentation:

2^9|a^2=fi,b^2=ei,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, 408485);