#10463929

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

4040 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10371125);