#199041

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

1023 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)(19,27)(20,32,34,24)(21,35)(22,26)(23,31)(25,28)(29,41,44,48)(30,42)(33,39,36,40)(37,45,47,46)(38,43);(2,7)(4,5)(8,11)(9,10)(12,13)(15,16)(17,19,28,43,22,30,21,31)(18,23,35,42,26,38,25,27)(20,29,33,45,34,44,36,46)(24,37,39,48,32,47,40,41);(1,3)(2,8)(4,12)(5,13)(6,14)(7,11)(9,15)(10,16)(17,20)(18,24)(19,29)(21,36)(22,34)(23,37)(25,40)(26,32)(27,41)(28,33)(30,44)(31,46)(35,39)(38,47)(42,48)(43,45);(1,4,6,5)(2,7,10,9)(3,12,14,13)(8,11,16,15)(19,30)(23,38)(27,42)(29,44)(31,43)(37,47)(41,48)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=i,g^2=i,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, 106237);