#10195883

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

4244 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,d^2=h,e^2=gh,f^2=ghi,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, 10103079);