#10368658

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

4087 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=i,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10275854);