#9136909

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	#53737	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4158 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 9044105);