#9159910

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	#51906	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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:

(1,2,7,9)(3,10,6,8)(4,13,14,5)(11,16,15,12)(17,18,22,23)(19,24,27,30)(20,25,31,28)(21,29,32,26)(33,34,40,37)(35,41,45,39)(36,38,42,47)(43,48,46,44);(1,3,7,6)(2,8,9,10)(4,11,14,15)(5,16,13,12)(17,18,22,23)(19,24,27,30)(20,28,31,25)(21,26,32,29)(33,35,40,45)(34,41,37,39)(36,46,42,43)(38,44,47,48);(2,9)(4,14)(8,10)(11,15)(18,23)(19,27)(25,28)(26,29)(33,36,40,42)(34,38,37,47)(35,43,45,46)(39,44,41,48);(1,4,7,14)(2,5,9,13)(3,11,6,15)(8,16,10,12)(17,19)(18,24)(20,29)(21,28)(22,27)(23,30)(25,32)(26,31)(33,37,40,34)(35,39,45,41)(36,38,42,47)(43,48,46,44);(1,5)(2,4)(3,12)(6,16)(7,13)(8,15)(9,14)(10,11)(17,20)(18,25)(19,26)(21,24)(22,31)(23,28)(27,29)(30,32)(33,38)(34,42)(35,44)(36,37)(39,46)(40,47)(41,43)(45,48);(1,6,7,3)(2,10,9,8)(4,15,14,11)(5,12,13,16)(17,21)(18,26)(19,28)(20,30)(22,32)(23,29)(24,31)(25,27)(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=ghi,d^2=hi,f^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, 9067106);