#8726434

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

4491 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8633630);