#8826247

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

4993 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,e^2=gh,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, 8733443);