#8994156

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

4302 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8901352);