#9712262

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	#48005	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4137 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,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, 9619458);