#199163

order	512
nilpotency class	3
isoclinism	isoclinic class of #198843
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,7,6,10,4,9)(3,11,12,8,14,15,13,16)(17,18,22,27,23,28,21,26)(19,29,31,39,33,41,32,24)(20,34,35,42,37,25,36,40)(30,43,46,38,45,48,44,47);(2,7)(4,5)(8,11)(9,10)(12,13)(15,16)(17,19)(18,24,28,39)(20,30)(21,32)(22,31)(23,33)(25,38,34,47)(26,41,27,29)(35,46)(36,44)(37,45)(40,48,42,43);(1,3)(2,8)(4,12)(5,13)(6,14)(7,11)(9,15)(10,16)(17,20,23,37)(18,25,28,34)(19,30,33,45)(21,35,22,36)(24,38,39,47)(26,40,27,42)(29,43,41,48)(31,44,32,46);(1,4,6,5)(2,7,10,9)(3,12,14,13)(8,11,16,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,39,41)(25,40,34,42)(30,44,45,46)(38,43,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=hi,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dhi,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,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, 106359);