#199847

order	512
nilpotency class	3
isoclinism	isoclinic class of #199842
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	#60803	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1209 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:

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

Polycyclic presentation:

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

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, 107043);