#8105100

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4163 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8012296);