#9851261

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

4141 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,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, 9758457);