#9712268

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	#48150	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 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)(3,11,12,7)(4,9,14,16)(5,15,13,10)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,34)(35,43)(36,46)(37,40)(38,42)(39,41)(44,47)(45,48);(1,3,6,12)(2,7,8,11)(4,13,14,5)(9,15,16,10)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35,40,39)(34,41,37,43)(36,45,38,44)(42,48,46,47);(2,8)(7,11)(9,16)(10,15)(17,18,23,21)(19,24,20,25)(22,26,30,29)(27,31,28,32)(33,36,40,38)(34,42,37,46)(35,44,39,45)(41,47,43,48);(1,2)(3,7)(4,9)(5,10)(6,8)(11,12)(13,15)(14,16)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,37,40,34)(35,43,39,41)(36,42,38,46)(44,47,45,48);(1,4,6,14)(2,9,8,16)(3,5,12,13)(7,10,11,15)(17,21,23,18)(19,25,20,24)(22,26,30,29)(27,31,28,32)(33,38,40,36)(34,42,37,46)(35,45,39,44)(41,47,43,48);(1,5)(2,10)(3,4)(6,13)(7,9)(8,15)(11,16)(12,14)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,39,40,35)(34,43,37,41)(36,45,38,44)(42,48,46,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,d^2=ghi,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,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, 9619464);