#419560

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1281 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:

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

Polycyclic presentation:

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

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, 326756);