#292233

order	512
nilpotency class	3
isoclinism	isoclinic class of #273051
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	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

777 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)(11,18)(12,16)(13,23)(14,17)(15,21)(19,24)(20,22)(25,26,30,33)(27,35,37,32)(28,29,34,31)(36,38,39,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,22)(13,20)(14,21)(16,24)(17,18)(19,23)(25,27)(26,32)(28,38)(29,36)(30,37)(31,39)(33,35)(34,40);(9,12,14,23)(10,16,17,13)(11,19)(15,24)(18,20)(21,22)(25,28,31,26,30,34,29,33)(27,32,36,38,37,35,39,40);(2,6)(5,7)(9,13,14,16)(10,12,17,23)(11,20,21,24)(15,22,18,19)(25,29,30,31)(26,28,33,34)(27,36,37,39)(32,38,35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=fhi,d^2=i,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=dhi,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 199429);