#8839560

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	#36842	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8746756);