#8096544

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

4394 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,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, 8003740);