#9713466

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	#48429	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,27)(19,26,22,24)(20,21,25,32)(28,29,30,31)(34,39)(35,42)(37,46)(44,47);(4,11)(8,14)(10,15)(12,16)(17,19,23,22)(18,24,27,26)(20,28,25,30)(21,31,32,29)(34,39)(36,41)(40,43)(44,47);(2,7)(3,9)(8,14)(10,15)(17,19,23,22)(18,24,27,26)(20,30,25,28)(21,29,32,31)(33,34,38,39)(35,43,42,40)(36,37,41,46)(44,48,47,45);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,25)(18,21,27,32)(19,28,22,30)(24,31,26,29)(33,35,38,42)(34,40,39,43)(36,44,41,47)(37,45,46,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,25)(19,29)(20,27)(22,31)(23,32)(24,28)(26,30)(33,36)(34,37)(35,44)(38,41)(39,46)(40,45)(42,47)(43,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22,23,19)(18,26,27,24)(20,30,25,28)(21,29,32,31)(33,37)(34,41)(35,45)(36,39)(38,46)(40,47)(42,48)(43,44)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^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, 9620662);