#8436254

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

4482 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)(9,10,15,18)(11,19,22,16)(12,23,21,14)(13,24,20,17)(27,33)(31,34)(35,39)(36,40);(3,7)(6,8)(9,11)(10,16)(12,20)(13,21)(14,17)(15,22)(18,19)(23,24)(25,26,30,32)(27,34,33,31)(28,37,38,29)(35,36,39,40);(1,2)(3,6)(4,5)(7,8)(25,27,30,33)(26,31,32,34)(28,35,38,39)(29,36,37,40);(1,3,4,7)(2,6,5,8)(9,12,15,21)(10,14,18,23)(11,20,22,13)(16,17,19,24)(25,28,30,38)(26,29,32,37)(27,35,33,39)(31,36,34,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,20)(10,17,18,24)(11,21,22,12)(14,19,23,16);(9,14)(10,12)(11,17)(13,19)(15,23)(16,20)(18,21)(22,24)(25,29)(26,28)(27,36)(30,37)(31,35)(32,38)(33,40)(34,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=gh,e^2=g,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, 8343450);