#198361

order	512
nilpotency class	3
isoclinism	isoclinic class of #198171
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3245	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1181 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=h,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105557);