#199912

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	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1002 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=hi,g^2=h,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, 107108);