#198123

order	512
nilpotency class	3
isoclinism	isoclinic class of #198117
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

998 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105319);