#303571

order	512
nilpotency class	3
isoclinism	isoclinic class of #271003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

910 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)(4,6)(5,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,31)(29,32);(1,3,5,4)(2,6,8,7)(10,15)(11,16)(17,19)(18,24)(20,21)(22,28)(23,29)(25,30)(26,31)(27,32);(1,3,5,4)(2,7,8,6)(9,11,12,10,13,16,14,15)(17,20,21,19,23,28,22,29)(18,24,26,30,27,32,25,31);(1,4,5,3)(2,7,8,6)(9,12,13,14)(10,16,15,11)(17,21,23,22)(18,25,27,26)(19,20,29,28)(24,30,32,31)

Polycyclic presentation:

2^9|b^2=ehi,c^2=ef,d^2=ei,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=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 210767);