#396381

order	512
nilpotency class	3
isoclinism	isoclinic class of #396381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14732
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1399 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303577);