#8944557

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

4702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,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, 8851753);