#9712484

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

4147 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)(3,7,11,14)(4,5,12,13)(9,10,15,16)(17,18,22,24)(19,23,26,21)(20,25,30,32)(27,29,28,31)(34,39)(35,42)(37,45)(44,47);(1,3,6,11)(2,7,8,14)(4,9,12,15)(5,10,13,16)(17,19,22,26)(18,23,24,21)(20,28,30,27)(25,31,32,29)(33,34,38,39)(35,36,42,40)(37,41,45,46)(43,44,48,47);(2,8)(4,12)(7,14)(9,15)(18,24)(19,26)(25,32)(27,28)(33,35,38,42)(34,40,39,36)(37,43,45,48)(41,47,46,44);(1,4)(2,5)(3,9)(6,12)(7,10)(8,13)(11,15)(14,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,29)(28,31)(30,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,20,22,30)(18,25,24,32)(19,27,26,28)(21,31,23,29)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,5)(2,4)(3,10)(6,13)(7,9)(8,12)(11,16)(14,15)(17,21,22,23)(18,26,24,19)(20,29,30,31)(25,27,32,28)(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=gi,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, 9619680);