#8271195

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

4176 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=ghi,e^2=hi,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, 8178391);