#320232

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
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

750 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)(3,7)(4,5)(6,8)(9,10,13,11,12,16,14,15)(17,18,22,24,20,25,21,26)(19,27,30,31,28,23,29,32);(2,5)(6,7)(9,11,13,16,12,15,14,10)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,27)(26,31);(10,15)(11,16)(13,14)(18,24)(19,28)(21,22)(23,31)(25,26)(27,32);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=fhi,b^2=gh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 227428);