#290111

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

575 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)(9,10,12,16,13,11,14,15)(17,18,21,30,22,20,23,25)(19,26,27,31,28,24,29,32);(1,3,5,4)(2,6,8,7)(9,11,12,15,13,10,14,16)(17,19,23,29,22,28,21,27)(18,24,25,31,20,26,30,32);(10,15)(11,16)(12,14)(17,20,22,18)(19,26,28,24)(21,30,23,25)(27,32,29,31);(1,4,5,3)(2,7,8,6)(17,21,22,23)(18,25,20,30)(19,27,28,29)(24,31,26,32)

Polycyclic presentation:

2^9|a^2=ef,b^2=efi,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, 197307);