#509866

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

503 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=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, 417062);