#9058795

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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)(3,11,12,8)(4,9,14,13)(5,15,16,10)(17,18)(19,23)(20,25)(21,29)(22,24)(26,30)(27,32)(28,31)(33,34,39,44)(35,45,48,40)(36,38,43,41)(37,42,46,47);(2,7)(3,12)(9,13)(10,15)(17,19,22,20)(18,23,24,25)(21,27,30,28)(26,31,29,32)(33,35,39,48)(34,40,44,45)(36,37,43,46)(38,47,41,42);(2,7)(3,12)(4,14)(5,16)(18,24)(19,20)(21,30)(31,32)(36,43)(37,46)(38,41)(42,47);(1,3,6,12)(2,8,7,11)(4,15,14,10)(5,13,16,9)(17,18)(19,25)(20,23)(21,26)(22,24)(27,32)(28,31)(29,30)(33,36,39,43)(34,41,44,38)(35,46,48,37)(40,47,45,42);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,16)(12,15)(17,20,22,19)(18,25,24,23)(21,28,30,27)(26,32,29,31)(33,37,39,46)(34,42,44,47)(35,36,48,43)(38,40,41,45);(1,5,6,16)(2,10,7,15)(3,13,12,9)(4,11,14,8)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,38,39,41)(34,43,44,36)(35,47,48,42)(37,40,46,45)

Polycyclic presentation:

2^9|a^2=g,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,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965991);