#506511

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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

842 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)(13,14,19,24,17,33,34,36,18,16,20,25,21,23,35,37)(15,26,30,41,28,39,42,27,29,22,31,40,32,43,38,44);(5,7,8,12)(6,9,10,11)(13,15,19,30,17,28,34,42,18,29,20,31,21,32,35,38)(14,22,24,40,33,43,36,44,16,26,25,41,23,39,37,27);(2,4)(13,16)(14,18)(15,27)(17,33)(19,36)(20,37)(21,23)(22,38)(24,34)(25,35)(26,42)(28,41)(29,44)(30,39)(31,43)(32,40);(6,10)(9,11)(13,17,18,21)(14,23,16,33)(15,28,29,32)(19,34,20,35)(22,39,26,43)(24,37,25,36)(27,41,44,40)(30,42,31,38)

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=de,c^b=cg,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413707);