#8810232

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

4321 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)(9,10,14,17)(11,12,15,20)(13,16,22,24)(18,19,21,23)(25,26)(27,33)(28,34)(29,32)(30,31)(35,39)(36,40)(37,38);(3,7)(6,8)(9,11,14,15)(10,12,17,20)(13,21,22,18)(16,23,24,19)(26,31)(27,34)(29,37)(36,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,18)(14,15)(16,19)(17,20)(21,22)(23,24)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,40)(38,39);(9,12)(10,15)(11,17)(13,19)(14,20)(16,21)(18,24)(22,23)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,21)(18,24)(22,23)(25,29,30,37)(26,32,31,38)(27,35,34,40)(28,36,33,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,22)(10,16,17,24)(11,18,15,21)(12,19,20,23)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717428);