#198517

order	512
nilpotency class	3
isoclinism	isoclinic class of #198516
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	C₂³×C₄	C₂⁴
prime radical	#60750	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1364 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi

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, 105713);