#206030

order	512
nilpotency class	3
isoclinism	isoclinic class of #205751
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1005 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)(3,6,10,14)(4,7)(5,8)(9,13,11,15)(12,16)(17,18,21,25,23,26,22,32)(19,20,27,29,28,30,31,24);(1,3)(2,6,8,14)(4,11)(5,10)(7,15,16,13)(9,12)(17,19,23,28)(18,24,26,29)(20,25,30,32)(21,31,22,27);(17,20,22,24,23,30,21,29)(18,19,25,27,26,28,32,31);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,g^b=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, 113226);