#499234

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂²×D₈
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

882 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)(7,10)(8,9)(11,12)(13,14)(15,20)(16,26)(17,29)(18,33)(19,23)(21,27)(22,31)(24,40)(25,39)(28,43)(30,35)(32,42)(34,38)(36,44)(37,41);(1,2)(3,4)(6,9)(10,12)(13,15,19,18)(14,20,23,33)(16,24,28,34)(17,25,32,31)(21,35,37,36)(22,29,39,42)(26,40,43,38)(27,30,41,44);(2,4)(13,16,19,28)(14,21,23,37)(15,24,18,34)(17,30,32,44)(20,35,33,36)(22,38,39,40)(25,41,31,27)(26,42,43,29);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406430);