#10519354

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

4238 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:

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

Polycyclic presentation:

2^9|a^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, 10426550);