#295981

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

769 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=hi,c^2=f,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,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 203177);