#9160821

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	#57954	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4145 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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18)(19,24)(20,28)(21,25)(22,26)(23,29)(27,31)(30,32)(33,34,39,43)(35,44,42,38)(36,47,46,40)(37,41,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19)(18,21)(20,27)(22,25)(23,31)(24,26)(28,32)(29,30)(33,35,39,42)(34,38,43,44)(36,45,46,37)(40,48,47,41);(2,6)(7,9)(8,12)(15,16)(20,29)(23,28)(27,30)(31,32)(33,36,39,46)(34,40,43,47)(35,45,42,37)(38,48,44,41);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,20,22,29)(18,23,26,28)(19,27,25,30)(21,31,24,32)(33,36)(34,40)(35,45)(37,42)(38,48)(39,46)(41,44)(43,47);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,19)(18,24)(20,30)(21,26)(22,25)(23,31)(27,29)(28,32)(33,37,39,45)(34,41,43,48)(35,46,42,36)(38,47,44,40);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,21,22,24)(18,25,26,19)(20,31,29,32)(23,30,28,27)(33,38)(34,42)(35,43)(36,48)(37,40)(39,44)(41,46)(45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=ghi,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, 9068017);