#268357

order	512
nilpotency class	3
isoclinism	isoclinic class of #268355
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,f^c=fhi,e^d=ei,f^d=fhi

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, 175553);