#243390

order	512
nilpotency class	3
isoclinism	isoclinic class of #233152
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

562 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)(9,10,15,20,14,19,13,18)(11,21,22,17,12,16,23,24)(25,26)(27,30)(28,31)(29,32)(33,40)(34,38)(35,37)(36,39);(2,5)(6,8)(9,11)(10,16)(12,14)(13,23)(15,22)(17,18)(19,21)(20,24)(25,27,28,34)(26,30,31,38)(29,33,36,35)(32,40,39,37);(9,12,14,11)(10,17,19,24)(13,23,15,22)(16,20,21,18)(26,31)(27,33)(29,36)(30,37)(34,35)(38,40);(1,3,4,7)(2,6,5,8)(9,13,14,15)(10,18,19,20)(11,22,12,23)(16,24,21,17)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150586);