#251457

order	512
nilpotency class	3
isoclinism	isoclinic class of #246799
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	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

828 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)(9,10,14,18)(11,19,21,12)(13,24,15,17)(16,23,20,22)(25,26,30,35)(27,37,28,32)(29,36,31,34)(33,38,40,39);(1,3,4,8)(2,5,6,7)(9,11,14,21)(10,12,18,19)(13,20,15,16)(17,22,24,23)(25,27,30,28)(26,32,35,37)(29,38,31,39)(33,34,40,36);(9,12,14,19)(10,16,18,20)(11,17,21,24)(13,23,15,22)(25,28,30,27)(26,33,35,40)(29,39,31,38)(32,36,37,34);(2,6)(5,7)(9,13,14,15)(10,17,18,24)(11,20,21,16)(12,22,19,23)(25,29,30,31)(26,34,35,36)(27,38,28,39)(32,40,37,33)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=h,d^2=h,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=dhi,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158653);