#509096

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

702 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)(7,11)(8,10)(9,12)(13,14,18,23)(15,25,20,21)(16,29,17,31)(19,34,33,39)(22,32,40,30)(24,28,41,37)(26,44,27,42)(35,43,36,38);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,20)(14,21,23,25)(16,26,17,27)(19,28,33,37)(22,38,40,43)(24,39,41,34)(29,42,31,44)(30,35,32,36);(2,4)(13,16,18,17)(14,22,23,40)(15,26,20,27)(19,35,33,36)(21,38,25,43)(24,42,41,44)(28,32,37,30)(29,39,31,34);(6,10)(7,12)(13,17,18,16)(14,22,23,40)(15,27,20,26)(19,36,33,35)(21,38,25,43)(24,42,41,44)(28,30,37,32)(29,39,31,34)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=dei,f^c=fh,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416292);