#416278

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323474);