#310146

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

653 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)(3,6)(4,5)(7,8)(9,10,11,15,12,14,16,13)(17,18,22,21)(19,23,26,29)(20,24,28,30)(25,31,32,27);(10,13)(11,16)(14,15)(17,19,22,26)(18,23,21,29)(20,27,28,31)(24,25,30,32);(2,5)(6,8)(10,14)(13,15)(17,20,22,28)(18,24,21,30)(19,25,26,32)(23,31,29,27);(1,3,4,7)(2,6,5,8)(10,13)(11,16)(14,15)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 217342);