#8534586

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

4126 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=h,e^2=i,f^2=g,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, 8441782);