#10168349

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

4048 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10075545);