#9711943

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

4099 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)(3,8)(4,6)(5,10)(7,9)(11,13)(12,15)(14,16)(18,23)(19,25)(26,31)(27,32)(33,34,39,37)(35,42,41,38)(36,44,46,40)(43,45,47,48);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,15,16,13)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31)(33,35,39,41)(34,38,37,42)(36,43,46,47)(40,45,44,48);(2,9)(3,4)(10,16)(11,12)(33,36,39,46)(34,40,37,44)(35,43,41,47)(38,45,42,48);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,15,16,13)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,29,31,28)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,5)(2,10)(3,11)(4,12)(6,15)(7,14)(8,13)(9,16)(17,20,22,24)(18,25,23,19)(21,29,30,28)(26,27,31,32)(33,37,39,34)(35,38,41,42)(36,44,46,40)(43,45,47,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,12)(11,16)(14,15)(17,21,22,30)(18,26,23,31)(19,27,25,32)(20,28,24,29)(33,38)(34,41)(35,37)(36,45)(39,42)(40,47)(43,44)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,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, 9619139);