#10247054

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

3989 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154250);