#10247477

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

4077 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)(3,6)(4,7)(5,10)(8,14)(9,11)(12,16)(13,15)(17,18,23,27)(19,28,21,24)(20,29,31,25)(22,26,32,30)(33,34,39,36)(35,37,40,43)(38,44,47,41)(42,46,48,45);(2,3)(4,11)(8,10)(12,13)(17,19,23,21)(18,24,27,28)(20,30,31,26)(22,29,32,25)(33,35)(34,37)(36,43)(38,42)(39,40)(41,45)(44,46)(47,48);(2,3)(7,9)(8,10)(15,16)(34,36)(35,40)(41,44)(42,48);(1,3,6,2)(4,7,11,9)(5,10,14,8)(12,15,13,16)(17,20,23,31)(18,25,27,29)(19,26,21,30)(22,24,32,28)(33,36,39,34)(35,37,40,43)(38,44,47,41)(42,45,48,46);(1,4,6,11)(2,7,3,9)(5,13,14,12)(8,16,10,15)(17,21,23,19)(18,24,27,28)(20,26,31,30)(22,29,32,25)(33,37)(34,40)(35,36)(38,46)(39,43)(41,42)(44,48)(45,47);(1,5,6,14)(2,8,3,10)(4,12,11,13)(7,15,9,16)(17,22)(18,26)(19,29)(20,24)(21,25)(23,32)(27,30)(28,31)(33,38,39,47)(34,41,36,44)(35,42,40,48)(37,45,43,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=gi,e^2=g,f^2=h,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, 10154673);