#8873767

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780963);