#259093

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,f^c=fh,g^c=gi

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, 166289);