#10247444

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

4251 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10154640);