#252478

order	512
nilpotency class	3
isoclinism	isoclinic class of #246239
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

915 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)(3,7)(4,6)(5,8)(9,10,13,17)(11,18,20,23)(12,22,14,16)(15,19,24,21)(25,26,28,31)(27,35,37,40)(29,34,30,33)(32,36,39,38);(1,3,4,8)(2,5,6,7)(9,10)(11,18)(12,16)(13,17)(14,22)(15,19)(20,23)(21,24)(26,31)(27,36)(29,30)(32,35)(37,38)(39,40);(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,24)(18,22)(25,27,28,37)(26,32,31,39)(29,36,30,38)(33,40,34,35);(2,6)(5,7)(9,12,13,14)(10,16,17,22)(11,19,20,21)(15,23,24,18)(26,31)(27,37)(33,34)(36,38)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=i,d^2=hi,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=di,f^c=fh,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159674);