#251415

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

843 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158611);