#198739

order	512
nilpotency class	3
isoclinism	isoclinic class of #198510
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dh,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, 105935);