#504652

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	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

559 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=e,d^2=f,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, 411848);