#8105136

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

4155 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=gi,e^2=g,f^2=h,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, 8012332);