#251276

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	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

885 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)(11,17)(12,14)(13,21)(15,19)(16,20)(18,24)(22,23)(25,26)(27,32)(28,37)(29,34)(30,39)(31,35)(33,38)(36,40);(1,3,4,8)(2,5,6,7)(10,14)(15,23)(16,21)(17,24)(25,27,31,28)(26,32,35,37)(29,36,30,38)(33,34,40,39);(9,11)(10,15)(12,18)(13,22)(14,23)(16,24)(17,21)(19,20)(25,28,31,27)(26,33,35,40)(29,38,30,36)(32,39,37,34);(2,6)(5,7)(10,14)(11,18)(16,21)(19,22)(25,29,31,30)(26,34,35,39)(27,36,28,38)(32,40,37,33)

Polycyclic presentation:

2^9|b^2=eh,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, 158472);