#199209

order	512
nilpotency class	3
isoclinism	isoclinic class of #199203
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:

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

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,d^b=dh,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, 106405);