#9947523

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

4202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9854719);