#349571

order	512
nilpotency class	3
isoclinism	isoclinic class of #346740
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

919 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,d^b=dh,f^c=fh,g^c=gi,f^d=fi,g^d=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, 256767);