#9127537

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
prime radical	#59779	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4094 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9034733);