#8701589

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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)(3,4,9,8)(5,10,15,16)(11,12,13,14)(17,18,23,21)(19,24,30,27)(20,31,29,25)(22,26,28,32)(33,34,39,38)(35,40,43,45)(36,41,44,47)(37,42,46,48);(2,7)(3,9)(10,16)(11,13)(17,19)(18,24)(20,28)(21,27)(22,29)(23,30)(25,26)(31,32)(35,43)(36,44)(40,45)(41,47);(17,20)(18,25)(19,22)(21,31)(23,29)(24,32)(26,27)(28,30)(33,35,39,43)(34,40,38,45)(36,37,44,46)(41,42,47,48);(1,3,6,9)(2,8,7,4)(5,13,15,11)(10,12,16,14)(17,21,23,18)(19,27,30,24)(20,31,29,25)(22,26,28,32)(33,36,39,44)(34,41,38,47)(35,37,43,46)(40,42,45,48);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,16,13)(17,20)(18,25)(19,28)(21,31)(22,30)(23,29)(24,26)(27,32)(33,37)(34,42)(35,44)(36,43)(38,48)(39,46)(40,47)(41,45);(1,5,6,15)(2,10,7,16)(3,11,9,13)(4,12,8,14)(17,22,23,28)(18,26,21,32)(19,29,30,20)(24,25,27,31)(33,38,39,34)(35,45,43,40)(36,41,44,47)(37,42,46,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,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, 8608785);