#399170

order	512
nilpotency class	3
isoclinism	isoclinic class of #393893
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	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

1508 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306366);