#8963203

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

4048 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)(3,4)(5,13)(6,7)(8,14)(9,12)(10,16)(11,15)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,34,39,37)(35,41,36,40)(38,45,46,42)(43,47,44,48);(1,3,6,9)(2,4,7,12)(5,10,14,15)(8,11,13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(34,37)(40,41)(42,45)(47,48);(2,7)(3,9)(5,14)(11,16)(19,25)(21,29)(23,26)(24,28)(33,35,39,36)(34,40,37,41)(38,44,46,43)(42,48,45,47);(1,4,6,12)(2,3,7,9)(5,11,14,16)(8,10,13,15)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,3)(2,4)(5,10)(6,9)(7,12)(8,11)(13,16)(14,15)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,37,39,34)(35,40,36,41)(38,45,46,42)(43,47,44,48);(1,5,6,14)(2,8,7,13)(3,10,9,15)(4,11,12,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

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

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, 8870399);