#295217

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

730 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)(4,5)(6,8)(9,10)(11,16)(12,21)(13,22)(14,17)(15,18)(19,23)(20,24)(25,26,29,32)(27,34,36,31)(28,38,33,30)(35,37,39,40);(2,5)(6,7)(9,11,15,20)(10,16,18,24)(12,13,19,14)(17,21,22,23)(25,27,29,36)(26,31,32,34)(28,39,33,35)(30,40,38,37);(9,12,13,11,15,19,14,20)(10,16,17,21,18,24,22,23)(25,28,30,26,29,33,38,32)(27,35,37,31,36,39,40,34);(1,3,4,8)(2,6,5,7)(26,32)(28,33)(31,34)(35,39)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=ei,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,d^c=di,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 202413);