#198305

order	512
nilpotency class	3
isoclinism	isoclinic class of #198135
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1001 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,d^2=h,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105501);