#8942274

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

4147 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8849470);