#8533118

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

4356 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=h,e^2=gh,f^2=g,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, 8440314);