#318789

order	512
nilpotency class	3
isoclinism	isoclinic class of #262731
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

558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225985);