#350827

order	512
nilpotency class	3
isoclinism	isoclinic class of #346644
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×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

894 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)(3,6)(4,5)(7,8)(9,10,13,17)(11,14,18,22)(12,15,20,16)(19,21,24,23)(25,26,30,34)(27,35,31,32)(28,38,29,39)(33,36,40,37);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,23)(17,22)(20,24)(25,27,30,31)(26,32,34,35)(28,36,29,37)(33,39,40,38);(1,3,4,7)(2,6,5,8)(10,15)(12,20)(14,21)(16,17)(19,24)(22,23)(25,28,30,29)(26,33,34,40)(27,36,31,37)(32,39,35,38);(11,18)(14,22)(19,24)(21,23)(25,29,30,28)(26,33,34,40)(27,37,31,36)(32,39,35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,d^b=dhi,f^c=fh,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258023);