#9851268

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

3939 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)(3,8,14,5)(4,6,16,9)(11,15,12,13)(17,18)(19,26)(20,28)(21,24)(22,23)(25,30)(27,31)(29,32)(33,34)(35,46)(36,41)(37,47)(38,43)(39,44)(40,45)(42,48);(1,3,7,14)(2,8,10,5)(4,12,16,11)(6,13,9,15)(17,19,23,21)(18,24,22,26)(20,27,30,29)(25,31,28,32)(33,35)(34,36)(37,43)(38,42)(39,48)(40,41)(44,47)(45,46);(1,4,7,16)(2,9,10,6)(3,11,14,12)(5,15,8,13)(17,19,23,21)(18,24,22,26)(20,29,30,27)(25,32,28,31)(33,36,40,46)(34,41,45,35)(37,38,48,44)(39,47,43,42);(1,4)(2,6)(3,12)(5,15)(7,16)(8,13)(9,10)(11,14)(17,20,23,30)(18,25,22,28)(19,27,21,29)(24,31,26,32)(33,37,40,48)(34,42,45,47)(35,43,41,39)(36,38,46,44);(1,5)(2,3)(4,15)(6,12)(7,8)(9,11)(10,14)(13,16)(17,21,23,19)(18,24,22,26)(20,29,30,27)(25,31,28,32)(33,38)(34,43)(35,47)(36,48)(37,46)(39,45)(40,44)(41,42);(1,6)(2,4)(3,13)(5,11)(7,9)(8,12)(10,16)(14,15)(17,22,23,18)(19,26,21,24)(20,28,30,25)(27,32,29,31)(33,39,40,43)(34,44,45,38)(35,48,41,37)(36,47,46,42)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9758464);