#8136788

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	#58237	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) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8043984);