#509080

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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

708 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)(13,14,20,25)(15,21,19,38)(16,29,17,31)(18,33,37,44)(22,32,23,30)(24,28,43,36)(26,42,27,41)(34,39,35,40);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,20,19)(14,21,25,38)(16,26,17,27)(18,28,37,36)(22,39,23,40)(24,33,43,44)(29,42,31,41)(30,34,32,35);(2,4)(13,16,20,17)(14,22,25,23)(15,26,19,27)(18,34,37,35)(21,39,38,40)(24,41,43,42)(28,32,36,30)(29,44,31,33);(6,10)(7,12)(13,17,20,16)(14,23,25,22)(15,27,19,26)(18,35,37,34)(21,40,38,39)(24,42,43,41)(28,30,36,32)(29,33,31,44)

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=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, 416276);