#9506840

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
prime radical	#47088	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3876 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18)(19,22)(20,24)(21,23)(25,26)(27,30)(28,29)(31,32)(33,34)(35,36)(37,42)(38,43)(39,40)(41,48)(44,46)(45,47);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,32)(29,31)(34,40)(35,41)(42,43)(44,45);(2,7)(6,13)(8,11)(14,15)(18,23)(19,26)(20,28)(30,31)(33,35,39,41)(34,36,40,48)(37,44,38,45)(42,46,43,47);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20,21,28)(18,24,23,29)(19,27,26,32)(22,30,25,31)(33,36)(34,41)(35,40)(37,46)(38,47)(39,48)(42,45)(43,44);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,18)(19,25)(20,29)(21,23)(22,26)(24,28)(27,30)(31,32)(33,37,39,38)(34,42,40,43)(35,44,41,45)(36,46,48,47);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,19)(18,25)(20,27)(21,26)(22,23)(24,31)(28,32)(29,30)(33,38,39,37)(34,43,40,42)(35,45,41,44)(36,47,48,46)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9414036);