#8097725

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

4582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004921);