#9712463

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	#49917	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3968 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)(4,9,13,16)(5,12,15,10)(17,18)(19,25)(20,24)(21,23)(22,26)(27,31)(28,30)(29,32)(36,44)(39,46)(42,48)(45,47);(1,3,6,11)(2,7,8,14)(4,12,13,10)(5,9,15,16)(17,19,22,21)(18,23,26,25)(20,28,29,27)(24,31,32,30)(33,34)(35,40)(36,39)(37,41)(38,43)(42,47)(44,46)(45,48);(2,8)(3,11)(9,16)(10,12)(17,18,22,26)(19,23,21,25)(20,24,29,32)(27,30,28,31)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,2)(3,7)(4,9)(5,10)(6,8)(11,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,47)(41,46)(43,48);(1,4,6,13)(2,9,8,16)(3,10,11,12)(5,14,15,7)(17,20,22,29)(18,24,26,32)(19,27,21,28)(23,30,25,31)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,48)(45,46);(1,5)(2,10)(3,9)(4,7)(6,15)(8,12)(11,16)(13,14)(17,21,22,19)(18,25,26,23)(20,27,29,28)(24,30,32,31)(33,34)(35,38)(36,39)(37,41)(40,43)(42,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,e^2=h,f^2=g,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, 9619659);