#266275

order	512
nilpotency class	3
isoclinism	isoclinic class of #247207
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×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

731 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)(11,15)(12,14)(13,16)(19,27)(20,21)(23,31)(24,25)(28,32)(29,30);(1,3,5,4)(2,6,8,7)(17,18,22,26)(19,28,29,31)(20,25,21,24)(23,27,32,30);(10,14)(11,12)(15,16)(17,19,22,29)(18,23,26,32)(20,30,21,27)(24,28,25,31);(1,4,5,3)(2,7,8,6)(9,11,13,12)(10,15,16,14)(19,29)(23,32)(27,30)(28,31)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=i,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 173471);