#9713470

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	#48405	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4131 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)(19,26)(20,23)(21,24)(22,28)(25,30)(27,31)(29,32)(33,34,40,37)(35,43,39,41)(36,45,38,42)(44,47,46,48);(4,11)(8,14)(10,15)(12,16)(19,21)(24,26)(27,29)(31,32)(33,35,40,39)(34,41,37,43)(36,44,38,46)(42,47,45,48);(2,7)(3,9)(8,14)(10,15)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,36,40,38)(34,42,37,45)(35,44,39,46)(41,47,43,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,37,40,34)(35,43,39,41)(36,42,38,45)(44,47,46,48);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,21,23,19)(18,24,20,26)(22,27,30,29)(25,32,28,31)(33,38,40,36)(34,42,37,45)(35,44,39,46)(41,48,43,47);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,39,40,35)(34,43,37,41)(36,46,38,44)(42,48,45,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=h,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, 9620666);