#8754838

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

4244 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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662034);