#9206763

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3896 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)(3,7)(4,5)(6,9)(8,12)(10,11)(13,16)(14,15)(17,18,24,27)(19,26,23,21)(20,29,22,25)(28,32,31,30)(34,38)(39,44)(40,45)(46,48);(1,3)(2,7)(4,10)(5,11)(6,12)(8,9)(13,14)(15,16)(17,19,24,23)(18,21,27,26)(20,28,22,31)(25,32,29,30)(36,43)(40,45)(41,47)(46,48);(1,3,6,12)(2,7,9,8)(4,10,14,13)(5,11,15,16)(17,20,24,22)(18,25,27,29)(19,28,23,31)(21,32,26,30)(33,34)(35,39)(36,40)(37,38)(41,46)(42,44)(43,45)(47,48);(1,3)(2,8)(4,13)(5,11)(6,12)(7,9)(10,14)(15,16)(17,21,24,26)(18,19,27,23)(20,30,22,32)(25,31,29,28)(33,35,37,42)(34,39,38,44)(36,41,43,47)(40,46,45,48);(1,4,6,14)(2,5,9,15)(3,10,12,13)(7,11,8,16)(17,22,24,20)(18,25,27,29)(19,28,23,31)(21,30,26,32)(33,36)(34,40)(35,41)(37,43)(38,45)(39,46)(42,47)(44,48);(1,5)(2,4)(3,11)(6,15)(7,10)(8,13)(9,14)(12,16)(17,23,24,19)(18,26,27,21)(20,31,22,28)(25,30,29,32)(33,34)(35,39)(36,40)(37,38)(41,46)(42,44)(43,45)(47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113959);