#9720530

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	#49727	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3987 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)(3,9)(4,10)(5,15)(6,7)(8,13)(11,16)(12,14)(17,18,22,27)(19,23,30,24)(20,25,26,21)(28,29,31,32)(36,44)(39,46)(43,47)(45,48);(2,7)(4,14)(8,9)(11,15)(17,19)(18,23)(20,31)(21,29)(22,30)(24,27)(25,32)(26,28)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(1,3,6,13)(2,8,7,9)(4,11,14,15)(5,12,16,10)(17,19,22,30)(18,24,27,23)(20,28,26,31)(21,29,25,32)(33,35,38,40)(34,37,41,42)(36,43,44,47)(39,45,46,48);(1,3)(2,9)(4,11)(5,12)(6,13)(7,8)(10,16)(14,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,27)(25,32)(26,31)(33,36)(34,39)(35,43)(37,45)(38,44)(40,47)(41,46)(42,48);(1,4,6,14)(2,10,7,12)(3,11,13,15)(5,9,16,8)(17,20)(18,25)(19,28)(21,27)(22,26)(23,29)(24,32)(30,31)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,46)(47,48);(1,5,6,16)(2,11,7,15)(3,12,13,10)(4,8,14,9)(17,21,22,25)(18,26,27,20)(19,29,30,32)(23,28,24,31)(33,37)(34,40)(35,41)(36,45)(38,42)(39,47)(43,46)(44,48)

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,e^2=g,f^2=h,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, 9627726);