#8105293

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

4139 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=g,e^2=i,f^2=gh,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, 8012489);