#506525

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	C₂×#2186	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

807 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)(13,14,17,22,16,30,31,40,19,23,24,34,18,21,33,32)(15,20,27,37,26,43,44,42,29,38,39,41,28,36,35,25);(5,7,8,12)(6,9,10,11)(13,15,24,39,16,26,33,35,19,29,17,27,18,28,31,44)(14,20,34,41,30,43,32,25,23,38,22,37,21,36,40,42);(2,4)(13,14)(15,25)(16,21)(17,32)(18,30)(19,23)(20,35)(22,33)(24,40)(26,41)(27,36)(28,37)(29,42)(31,34)(38,44)(39,43);(6,10)(9,11)(13,16,19,18)(14,21,23,30)(15,26,29,28)(17,31,24,33)(20,36,38,43)(22,32,34,40)(25,41,42,37)(27,44,39,35)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,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, 413721);