#9160820

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3933 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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18)(19,28)(20,31)(21,29)(22,25)(23,27)(24,30)(26,32)(33,34,40,37)(35,41,45,39)(36,38,42,47)(43,48,46,44);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19)(18,24)(20,29)(21,25)(22,26)(23,30)(27,28)(31,32)(33,35,40,45)(34,41,37,39)(36,46,42,43)(38,44,47,48);(2,6)(7,9)(8,12)(15,16)(17,20,23,22)(18,25,27,31)(19,29,30,26)(21,28,32,24)(33,36,40,42)(34,38,37,47)(35,43,45,46)(39,44,41,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,21,23,32)(18,26,27,29)(19,25,30,31)(20,24,22,28)(33,37,40,34)(35,39,45,41)(36,38,42,47)(43,48,46,44);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,22,23,20)(18,25,27,31)(19,29,30,26)(21,24,32,28)(33,38)(34,42)(35,44)(36,37)(39,46)(40,47)(41,43)(45,48);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,19)(18,24)(20,29)(21,25)(22,26)(23,30)(27,28)(31,32)(33,39)(34,35)(36,44)(37,45)(38,43)(40,41)(42,48)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=hi,e^2=h,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, 9068016);