#8701591

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

3941 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,18,23,22)(19,24,27,29)(20,25,28,31)(21,26,30,32)(33,34,40,39)(35,45,47,41)(36,48,46,42)(37,44,38,43);(2,3)(7,9)(8,10)(15,16)(19,27)(20,28)(24,29)(25,31)(33,35)(34,41)(36,37)(38,46)(39,45)(40,47)(42,44)(43,48);(17,19,23,27)(18,24,22,29)(20,21,28,30)(25,26,31,32)(33,36,40,46)(34,42,39,48)(35,37,47,38)(41,44,45,43);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20,23,28)(18,25,22,31)(19,21,27,30)(24,26,29,32)(33,37,40,38)(34,43,39,44)(35,36,47,46)(41,48,45,42);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21)(18,26)(19,28)(20,27)(22,32)(23,30)(24,31)(25,29)(33,38,40,37)(34,44,39,43)(35,46,47,36)(41,42,45,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22,23,18)(19,29,27,24)(20,25,28,31)(21,26,30,32)(33,39,40,34)(35,45,47,41)(36,48,46,42)(37,43,38,44)

Polycyclic presentation:

2^9|a^2=hi,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, 8608787);