#330834

order	512
nilpotency class	3
isoclinism	isoclinic class of #329551
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

516 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:

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

Polycyclic presentation:

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

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, 238030);