#291528

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×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

834 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)(11,16)(12,13)(14,21)(15,17)(18,19)(20,24)(22,23)(25,26)(27,30)(28,31)(29,32)(33,40)(34,38)(35,37)(36,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,18)(13,19)(14,22)(15,20)(17,24)(21,23)(25,27,28,34)(26,30,31,38)(29,33,36,35)(32,40,39,37);(9,12,13,10,15,21,14,17)(11,18,22,24,20,23,19,16)(26,31)(27,33)(29,36)(30,37)(34,35)(38,40);(2,6)(5,7)(9,13,15,14)(10,12,17,21)(11,19,20,22)(16,18,24,23)

Polycyclic presentation:

2^9|b^2=ei,c^2=fhi,d^2=h,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, 198724);