#10196269

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

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

4116 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10103465);