#8534566

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

4340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8441762);