#506516

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
prime radical	C₂³×D₈	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

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,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, 413712);