#317667

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

548 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=gh,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, 224863);