#10349042

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

4107 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)(4,8,12,13)(11,14,15,16)(17,18)(19,27)(20,29)(21,26)(22,24)(23,30)(25,31)(28,32)(33,34)(35,44)(36,40)(37,41)(38,45)(39,43)(42,48)(46,47);(2,6)(4,12)(7,10)(11,15)(17,19,22,30)(18,23,24,27)(20,26,31,28)(21,29,32,25)(35,36)(38,48)(40,44)(42,45);(3,9)(7,10)(11,15)(14,16)(18,24)(21,32)(23,27)(25,29)(33,35,39,36)(34,40,43,44)(37,42,47,45)(38,46,48,41);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20,22,31)(18,25,24,29)(19,28,30,26)(21,23,32,27)(33,36,39,35)(34,40,43,44)(37,42,47,45)(38,41,48,46);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,21,22,32)(18,26,24,28)(19,29,30,25)(20,23,31,27)(33,37,39,47)(34,41,43,46)(35,45,36,42)(38,40,48,44);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(33,38,39,48)(34,42,43,45)(35,46,36,41)(37,40,47,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=i,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10256238);