#10159199

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

4037 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10066395);