#10519356

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

4061 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=hi,e^2=hi,f^2=gh,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, 10426552);