#8096796

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

4540 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=g,e^2=gh,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, 8003992);