#326920

order	512
nilpotency class	3
isoclinism	isoclinic class of #326869
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

843 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)(3,6)(4,5)(7,8)(9,10,13,18)(11,19,22,15)(12,16,14,17)(20,24,21,23)(25,26,30,33)(27,31,28,36)(29,38,39,34)(32,35,40,37);(2,5)(6,8)(9,11,13,22)(10,15,18,19)(12,21,14,20)(16,24,17,23)(25,27,30,28)(26,31,33,36)(29,37,39,35)(32,34,40,38);(1,3)(2,6)(4,7)(5,8)(10,16)(11,20)(12,14)(17,18)(21,22)(23,24)(25,28,30,27)(26,32,33,40)(29,35,39,37)(31,38,36,34);(3,7)(6,8)(9,12,13,14)(10,17,18,16)(11,21,22,20)(15,23,19,24)(26,33)(31,36)(32,40)(34,38)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234116);