#251368

order	512
nilpotency class	3
isoclinism	isoclinic class of #249199
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

863 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)(11,20)(12,19)(13,18)(14,17)(15,22)(16,21)(23,24)(25,26)(27,31)(28,36)(29,38)(30,34)(32,37)(33,39)(35,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,23)(13,22)(14,21)(16,19)(17,24)(18,20)(25,27,30,28)(26,31,34,36)(29,37,39,35)(32,33,40,38);(9,11)(10,16)(12,21)(13,22)(14,23)(15,19)(17,20)(18,24)(25,28,30,27)(26,32,34,40)(29,35,39,37)(31,38,36,33);(2,6)(5,7)(9,12,13,14)(10,17,18,19)(11,21,22,23)(15,16,20,24)(27,28)(31,36)(32,40)(35,37)

Polycyclic presentation:

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

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, 158564);