#8112571

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
prime radical	#40139	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=i,e^2=g,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^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8019767);