#282317

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=ghi

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, 189513);