#8946204

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

4506 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=g,e^2=gh,f^2=gh,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, 8853400);