#9206568

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	#40332	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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)(3,8)(4,7)(5,9)(6,12)(10,15)(11,13)(14,16)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)(33,34,38,42)(35,43,44,39)(36,40,46,48)(37,41,45,47);(1,3,7,9)(2,8,4,5)(6,11,15,16)(10,14,12,13)(17,19)(18,22)(20,29)(21,30)(23,26)(24,25)(27,28)(31,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,45)(43,44)(46,48);(2,4)(3,9)(10,12)(11,16)(17,20,23,30)(18,24,27,31)(19,29,26,21)(22,25,28,32)(33,35,38,44)(34,39,42,43)(36,45,46,37)(40,41,48,47);(1,4,7,2)(3,5,9,8)(6,12,15,10)(11,13,16,14)(17,20)(18,24)(19,29)(21,26)(22,25)(23,30)(27,31)(28,32)(33,36,38,46)(34,40,42,48)(35,37,44,45)(39,47,43,41);(1,5)(2,9)(3,4)(6,14)(7,8)(10,11)(12,16)(13,15)(17,21,23,29)(18,25,27,32)(19,30,26,20)(22,24,28,31)(33,37)(34,41)(35,36)(38,45)(39,48)(40,43)(42,47)(44,46);(1,6,7,15)(2,10,4,12)(3,11,9,16)(5,13,8,14)(17,22,23,28)(18,26,27,19)(20,25,30,32)(21,31,29,24)(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=i,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, 9113764);