#8270525

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

4229 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,14)(15,16)(18,21)(19,20)(26,29)(27,28);(2,7)(4,8)(9,14)(12,15)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,34,39,38)(35,42,36,40)(37,45,46,41)(43,47,44,48);(5,13)(9,14)(10,16)(12,15)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35,39,36)(34,40,38,42)(37,43,46,44)(41,47,45,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,36,39,35)(34,40,38,42)(37,44,46,43)(41,47,45,48);(1,4,6,8)(2,3,7,11)(5,12,13,15)(9,10,14,16)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31)(33,37,39,46)(34,41,38,45)(35,43,36,44)(40,47,42,48);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,38,39,34)(35,40,36,42)(37,45,46,41)(43,47,44,48)

Polycyclic presentation:

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

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, 8177721);