#9863850

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

4123 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)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,18)(19,23)(20,21)(22,24)(25,29)(26,27)(28,30)(31,32)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,19,22,28)(18,23,24,30)(20,26,25,31)(21,27,29,32)(34,40)(35,37)(42,46)(43,45);(2,6)(7,14)(8,11)(15,16)(18,24)(20,25)(23,30)(26,31)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,20,22,25)(18,21,24,29)(19,26,28,31)(23,27,30,32)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,4)(2,8)(3,9)(5,13)(6,11)(7,15)(10,12)(14,16)(17,21)(18,25)(19,27)(20,24)(22,29)(23,31)(26,30)(28,32)(33,38,39,47)(34,42,40,46)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9771046);