#287225

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

611 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)(9,10,12,16,13,11,14,15)(17,18,22,29,21,25,20,24)(19,23,30,32,28,26,27,31);(2,5)(6,7)(9,11,12,15,13,10,14,16)(17,19,20,27,21,28,22,30)(18,23,24,31,25,26,29,32);(10,15)(11,16)(12,14)(17,18,21,25)(19,26,28,23)(20,29,22,24)(27,31,30,32);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=i,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 194421);