#243518

order	512
nilpotency class	3
isoclinism	isoclinic class of #233245
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

569 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)(9,10,14,16,13,17,12,18)(11,15,21,22,19,24,20,23);(2,5)(6,8)(9,11)(10,15)(12,20)(13,19)(14,21)(16,22)(17,24)(18,23)(25,26,29,32)(27,33,35,40)(28,36,38,31)(30,37,39,34);(10,16)(12,14)(15,22)(17,18)(20,21)(23,24)(25,27,29,35)(26,30,32,39)(28,37,38,34)(31,40,36,33);(1,3,4,7)(2,6,5,8)(10,17)(11,19)(16,18)(20,21)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=i,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,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150714);