#404484

order	512
nilpotency class	3
isoclinism	isoclinic class of #393893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=e,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^c=fi,h^c=hi,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, 311680);