#8119617

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

4132 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:

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

Polycyclic presentation:

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

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, 8026813);