#198571

order	512
nilpotency class	3
isoclinism	isoclinic class of #198522
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

982 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi

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, 105767);