#259130

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

761 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)(4,5)(6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,27)(28,36,39,40)(29,33,30,34)(32,38,35,37);(2,5)(6,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,21)(23,24)(25,27,31,26)(28,36,39,40)(29,34,30,33)(32,38,35,37);(11,18)(13,21)(14,22)(16,24)(17,23)(19,20)(25,28,31,39)(26,32,27,35)(29,38,30,37)(33,36,34,40);(1,3,4,8)(2,6,5,7)(10,15)(14,17)(16,24)(22,23)(25,29,31,30)(26,33,27,34)(28,37,39,38)(32,40,35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=ei,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,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 166326);