#8649956

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=gi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8557152);