#9712327

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 9619523);