#279481

order	512
nilpotency class	3
isoclinism	isoclinic class of #267211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

612 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,c^2=hi,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,f^c=fh,g^c=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, 186677);