#350461

order	512
nilpotency class	3
isoclinism	isoclinic class of #346346
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	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257657);