#8475318

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

4572 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8382514);