#201530

order	512
nilpotency class	3
isoclinism	isoclinic class of #9467
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1249 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,22,26,23,19,21,25)(20,27,28,32,29,24,30,31);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(17,19,22,25,23,18,21,26)(20,27,30,31,29,24,28,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,29)(25,31)(26,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,21,23,22)(18,25,19,26)(20,28,29,30)(24,31,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108726);