#8112566

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

4002 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,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, 8019762);