#9713381

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

3997 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)(3,13)(4,10)(5,11)(6,16)(7,9)(8,14)(12,15)(17,18)(19,27)(20,30)(21,24)(22,26)(23,25)(28,31)(29,32)(34,39)(35,42)(37,46)(44,47);(1,3,7,14)(2,8,9,13)(4,12,11,16)(5,15,10,6)(17,19,24,23)(18,25,21,27)(20,29,22,28)(26,32,30,31)(34,39)(36,41)(40,43)(44,47);(2,9)(4,11)(8,13)(12,16)(17,20,24,22)(18,26,21,30)(19,28,23,29)(25,31,27,32)(33,34,38,39)(35,43,42,40)(36,37,41,46)(44,48,47,45);(1,4,7,11)(2,10,9,5)(3,12,14,16)(6,13,15,8)(17,21,24,18)(19,27,23,25)(20,26,22,30)(28,31,29,32)(33,35,38,42)(34,40,39,43)(36,44,41,47)(37,45,46,48);(1,5,7,10)(2,11,9,4)(3,6,14,15)(8,12,13,16)(17,22,24,20)(18,26,21,30)(19,29,23,28)(25,31,27,32)(33,36)(34,37)(35,44)(38,41)(39,46)(40,45)(42,47)(43,48);(1,6)(2,12)(3,5)(4,13)(7,15)(8,11)(9,16)(10,14)(17,23,24,19)(18,27,21,25)(20,29,22,28)(26,32,30,31)(33,37)(34,41)(35,45)(36,39)(38,46)(40,47)(42,48)(43,44)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,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, 9620577);