#9206626

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	#49673	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4043 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)(3,6)(4,7)(5,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,30)(31,32)(33,34,38,42)(35,43,44,39)(36,40,46,48)(37,41,45,47);(1,3)(2,6)(4,10)(5,13)(7,9)(8,14)(11,15)(12,16)(17,19,23,22)(18,24,27,28)(20,29,26,31)(21,30,25,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,45)(43,44)(46,48);(2,4)(3,9)(8,12)(11,13)(17,20,23,26)(18,25,27,21)(19,29,22,31)(24,32,28,30)(33,35,38,44)(34,39,42,43)(36,45,46,37)(40,41,48,47);(1,4,7,2)(3,10,9,6)(5,8,15,12)(11,16,13,14)(17,20)(18,25)(19,29)(21,27)(22,31)(23,26)(24,32)(28,30)(33,36,38,46)(34,40,42,48)(35,37,44,45)(39,47,43,41);(1,5)(2,8)(3,11)(4,12)(6,16)(7,15)(9,13)(10,14)(17,21,23,25)(18,26,27,20)(19,30,22,32)(24,31,28,29)(33,37)(34,41)(35,36)(38,45)(39,48)(40,43)(42,47)(44,46);(1,6,7,10)(2,9,4,3)(5,14,15,16)(8,11,12,13)(17,22,23,19)(18,24,27,28)(20,31,26,29)(21,30,25,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,45)(43,44)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=g,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113822);