#506523

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

836 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)(13,14,17,23,16,32,33,36,20,25,18,24,19,22,34,35)(15,21,28,39,27,43,37,26,31,41,29,40,30,38,44,42);(5,7,8,12)(6,9,10,11)(13,15,18,29,16,27,34,44,20,31,17,28,19,30,33,37)(14,21,24,40,32,43,35,42,25,41,23,39,22,38,36,26);(2,4)(13,14)(15,26)(16,22)(17,35)(18,36)(19,32)(20,25)(21,37)(23,34)(24,33)(27,39)(28,43)(29,38)(30,40)(31,42)(41,44);(6,10)(9,11)(13,16,20,19)(14,22,25,32)(15,27,31,30)(17,33,18,34)(21,38,41,43)(23,35,24,36)(26,39,42,40)(28,37,29,44)

Polycyclic presentation:

2^9|a^2=f,b^2=eg,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,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, 413719);