#506537

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	C₂×#2218	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

808 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)(13,14,20,29,32,30,33,35,19,16,17,22,18,23,36,31)(15,21,34,43,41,42,37,40,25,28,26,38,27,39,44,24);(5,7,8,12)(6,9,10,11)(13,15,17,26,32,41,36,44,19,25,20,34,18,27,33,37)(14,21,22,38,30,42,31,24,16,28,29,43,23,39,35,40);(2,4)(13,16,19,14)(15,24,25,40)(17,31,20,35)(18,23,32,30)(21,37,28,44)(22,33,29,36)(26,42,34,39)(27,43,41,38);(6,10)(9,11)(15,25)(21,28)(24,40)(26,34)(27,41)(37,44)(38,43)(39,42)

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,c^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,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, 413733);