#297009

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
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

686 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)(11,14)(12,13)(15,16)(17,18)(19,24)(20,21)(22,29)(23,25)(26,30)(27,31)(28,32);(2,5)(6,7)(9,11,14,10)(12,13,16,15)(17,19,23,28)(18,24,25,32)(20,26,29,31)(21,30,22,27);(9,12,15,11,14,16,13,10)(17,20,21,18,23,29,22,25)(19,26,30,24,28,31,27,32);(1,3,4,8)(2,6,5,7)(9,13,14,15)(10,16,11,12)(17,21,23,22)(18,20,25,29)(19,27,28,30)(24,31,32,26)

Polycyclic presentation:

2^9|a^2=ei,b^2=h,c^2=fh,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,d^b=dh,f^b=fi,g^b=gi,d^c=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204205);