#199200

order	512
nilpotency class	3
isoclinism	isoclinic class of #199194
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	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

990 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=i,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,g^b=gi,d^c=dh,f^c=fi

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, 106396);