#9159902

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	#47704	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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)(3,8,11,14)(4,9,15,12)(5,13,16,10)(17,18,24,27)(19,28,31,25)(20,32,22,23)(21,29,30,26)(33,34,40,37)(35,41,45,39)(36,38,42,47)(43,48,46,44);(1,2,6,7)(3,8,11,14)(4,12,15,9)(5,10,16,13)(17,19,24,31)(18,25,27,28)(20,29,22,26)(21,23,30,32)(33,35,40,45)(34,41,37,39)(36,46,42,43)(38,44,47,48);(2,7)(3,11)(9,12)(10,13)(17,20,24,22)(18,23,27,32)(19,29,31,26)(21,25,30,28)(33,36,40,42)(34,38,37,47)(35,43,45,46)(39,44,41,48);(1,3)(2,8)(4,13)(5,12)(6,11)(7,14)(9,16)(10,15)(17,21,24,30)(18,26,27,29)(19,23,31,32)(20,28,22,25)(33,37,40,34)(35,39,45,41)(36,38,42,47)(43,48,46,44);(1,4)(2,9)(3,10)(5,8)(6,15)(7,12)(11,13)(14,16)(17,22,24,20)(18,23,27,32)(19,29,31,26)(21,28,30,25)(33,38)(34,42)(35,44)(36,37)(39,46)(40,47)(41,43)(45,48);(1,5)(2,10)(3,12)(4,14)(6,16)(7,13)(8,15)(9,11)(17,23)(18,22)(19,30)(20,27)(21,31)(24,32)(25,26)(28,29)(33,39)(34,35)(36,44)(37,45)(38,43)(40,41)(42,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9067098);