#8115457

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

3979 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022653);