#9236458

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

3954 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,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, 9143654);