#291737

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#59217	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

865 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,c^2=fhi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198933);