#8662933

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570129);