#506191

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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

705 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)(15,20)(16,21)(17,23)(18,24)(19,22)(25,33)(26,34)(27,35)(28,36)(29,37)(30,39)(31,38)(32,40)(41,43)(42,44);(5,7,8,12)(6,9,10,11)(13,15,25,28,31,29,32,41,19,16,17,26,18,27,42,30)(14,20,33,36,38,37,40,43,22,21,23,34,24,35,44,39);(2,4)(13,16,19,15)(14,21,22,20)(17,30,25,41)(18,27,31,29)(23,39,33,43)(24,35,38,37)(26,32,28,42)(34,40,36,44);(6,10)(9,11)(14,22)(20,21)(23,33)(24,38)(34,36)(35,37)(39,43)(40,44)

Polycyclic presentation:

2^9|b^2=egi,c^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413387);