#110783

order	512
nilpotency class	3
isoclinism	isoclinic class of #9519
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh

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, 17979);