#8533514

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

4652 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)(3,4)(9,10,14,15)(11,19,20,16)(12,22,23,17)(13,18,24,21)(25,26)(27,28)(29,33)(30,31)(32,34)(35,36)(37,40)(38,39);(1,2,6,5)(3,8,4,7)(10,15)(11,20)(17,22)(18,21)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,39)(38,40);(1,3,6,4)(2,7,5,8)(10,15)(12,23)(16,19)(18,21)(26,31)(27,34)(33,39)(35,40);(1,4,6,3)(2,8,5,7)(9,11,14,20)(10,16,15,19)(12,21,23,18)(13,22,24,17)(25,28,30,32)(26,27,31,34)(29,37,38,36)(33,40,39,35);(1,5,6,2)(3,8,4,7)(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,19,24,16)(25,27)(26,32)(28,31)(29,35)(30,34)(33,37)(36,39)(38,40);(9,13)(10,18)(11,17)(12,19)(14,24)(15,21)(16,23)(20,22)(25,29,30,38)(26,33,31,39)(27,35,34,40)(28,36,32,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

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, 8440710);