#324836

order	512
nilpotency class	3
isoclinism	isoclinic class of #278923
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

526 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)(17,18)(19,21)(20,27)(22,26)(23,29)(24,30)(25,31)(28,32);(2,5)(6,7)(9,10,13,14,12,16,11,15)(17,19,21,18)(20,23,28,25)(22,26,30,24)(27,31,32,29);(10,14)(11,13)(15,16)(17,20,22,29,21,28,30,31)(18,23,24,27,19,25,26,32);(1,3,4,8)(2,6,5,7)(18,19)(23,25)(24,26)(27,32)

Polycyclic presentation:

2^9|a^2=eh,b^2=gi,c^2=f,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,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, 232032);