#9236463

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	#50658	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4114 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9143659);