#328336

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, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

679 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=e,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, 235532);