#8433670

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

4624 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)(3,7)(9,10,16,15)(11,17,14,19)(12,22,13,18)(20,24,21,23)(28,36)(32,39)(34,40)(37,38);(1,2,4,5)(3,6,7,8)(9,11,16,14)(10,17,15,19)(12,21,13,20)(18,24,22,23)(25,26)(27,31)(28,32)(29,33)(30,35)(34,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,18,15,22)(11,20,14,21)(17,23,19,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,13,16,12)(10,18,15,22)(11,21,14,20)(17,23,19,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,14,16,11)(10,19,15,17)(12,21,13,20)(18,24,22,23)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40);(9,15,16,10)(11,19,14,17)(12,22,13,18)(20,24,21,23)(25,26)(27,30)(28,32)(29,33)(31,35)(34,37)(36,39)(38,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8340866);