#9712163

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	#42041	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4013 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)(3,8)(4,6)(5,10)(7,9)(11,13)(12,15)(14,16)(17,18,22,24)(19,27,28,23)(20,25,30,32)(21,31,29,26)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,15,16,13)(17,19,22,28)(18,23,24,27)(20,29,30,21)(25,31,32,26)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(2,9)(3,4)(10,16)(11,12)(18,24)(23,27)(25,32)(26,31)(33,34,39,37)(35,40,36,41)(38,42,46,45)(43,47,44,48);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,15,16,13)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,31)(30,32)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,5)(2,10)(3,11)(4,12)(6,15)(7,14)(8,13)(9,16)(17,20,22,30)(18,25,24,32)(19,21,28,29)(23,26,27,31)(33,37,39,34)(35,41,36,40)(38,42,46,45)(43,47,44,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,12)(11,16)(14,15)(17,21)(18,26)(19,20)(22,29)(23,25)(24,31)(27,32)(28,30)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=gh,e^2=g,f^2=ghi,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, 9619359);