#9712478

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4116 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)(3,7,10,5)(4,9,14,16)(11,13,12,15)(17,18)(19,20)(21,29)(22,26)(23,24)(25,31)(27,32)(28,30)(34,39)(35,42)(37,45)(44,47);(1,3,6,10)(2,7,8,5)(4,12,14,11)(9,15,16,13)(17,19,22,24)(18,23,26,20)(21,27,31,30)(25,32,29,28)(33,34,38,39)(35,36,42,40)(37,41,45,46)(43,44,48,47);(2,8)(3,10)(9,16)(11,12)(17,20,22,23)(18,24,26,19)(21,28,31,32)(25,30,29,27)(33,35,38,42)(34,40,39,36)(37,43,45,48)(41,47,46,44);(1,2)(3,7)(4,9)(5,10)(6,8)(11,13)(12,15)(14,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,4,6,14)(2,9,8,16)(3,11,10,12)(5,15,7,13)(17,19)(18,20)(21,30)(22,24)(23,26)(25,32)(27,31)(28,29)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,5,6,7)(2,10,8,3)(4,13,14,15)(9,11,16,12)(17,21)(18,25)(19,27)(20,28)(22,31)(23,32)(24,30)(26,29)(33,37,38,45)(34,41,39,46)(35,43,42,48)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9619674);