#8097291

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

4423 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=h,e^2=g,f^2=g,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, 8004487);