#8942397

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

4162 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=gh,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, 8849593);