#8119609

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

4124 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)(4,11,12,7)(10,16,15,14)(17,18)(19,24)(20,26)(21,29)(22,28)(23,25)(27,32)(30,31)(33,34,38,40)(35,42,37,39)(36,44,46,41)(43,47,45,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,19,23,21)(18,24,25,29)(20,27,30,28)(22,26,32,31)(33,35,38,37)(34,39,40,42)(36,45,46,43)(41,48,44,47);(3,9)(6,13)(10,15)(14,16)(18,25)(19,21)(26,31)(27,28)(34,40)(39,42)(41,44)(47,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20,23,30)(18,26,25,31)(19,28,21,27)(22,29,32,24)(33,36,38,46)(34,41,40,44)(35,43,37,45)(39,47,42,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,21,23,19)(18,24,25,29)(20,27,30,28)(22,31,32,26)(33,34)(35,39)(36,41)(37,42)(38,40)(43,47)(44,46)(45,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,22)(18,27)(19,26)(20,29)(21,31)(23,32)(24,30)(25,28)(33,37,38,35)(34,42,40,39)(36,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,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, 8026805);