#10188278

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

4078 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gh,e^2=g,f^2=hi,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, 10095474);