#10506324

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

4084 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)(3,8,11,13)(4,9,14,16)(5,10,15,12)(17,18)(19,26)(20,24)(21,29)(22,23)(25,30)(27,31)(28,32)(34,39)(35,40)(37,45)(43,46);(2,7)(3,11)(9,16)(10,12)(18,23)(24,26)(25,29)(31,32)(34,39)(35,40)(41,47)(42,48);(17,19,22,20)(18,24,23,26)(21,27,30,28)(25,31,29,32)(33,34,38,39)(35,36,40,44)(37,41,45,47)(42,43,48,46);(1,3)(2,8)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,18)(19,24)(20,26)(21,25)(22,23)(27,31)(28,32)(29,30)(33,35)(34,36)(37,42)(38,40)(39,44)(41,43)(45,48)(46,47);(1,4)(2,9)(3,12)(5,8)(6,14)(7,16)(10,11)(13,15)(17,20,22,19)(18,24,23,26)(21,27,30,28)(25,32,29,31)(33,36)(34,40)(35,39)(37,46)(38,44)(41,42)(43,45)(47,48);(1,5)(2,10)(3,9)(4,13)(6,15)(7,12)(8,14)(11,16)(17,21,22,30)(18,25,23,29)(19,27,20,28)(24,31,26,32)(33,37,38,45)(34,41,39,47)(35,42,40,48)(36,43,44,46)

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,e^2=hi,f^2=h,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, 10413520);