#251266

order	512
nilpotency class	3
isoclinism	isoclinic class of #249139
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

866 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)(9,10,13,16)(11,17,18,24)(12,20,22,15)(14,21,23,19)(25,26,30,35)(27,37,28,32)(29,36,31,34)(33,38,40,39);(1,3,4,8)(2,5,6,7)(25,27,30,28)(26,32,35,37)(29,38,31,39)(33,34,40,36);(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,20)(19,22)(25,28,30,27)(26,33,35,40)(29,39,31,38)(32,36,37,34);(2,6)(5,7)(11,18)(14,23)(17,24)(19,21)(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=i,d^2=i,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,d^c=dhi,f^c=fh

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, 158462);