#8517364

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

4512 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)(9,10,14,15)(11,19,20,16)(12,22,23,17)(13,18,24,21)(25,26)(27,31)(28,33)(29,35)(30,32)(34,36)(37,39)(38,40);(1,2,4,6)(3,5,7,8)(10,15)(11,20)(17,22)(18,21)(25,27)(26,31)(28,35)(29,33)(30,36)(32,34)(37,38)(39,40);(1,3)(2,5)(4,7)(6,8)(10,15)(12,23)(16,19)(18,21)(26,32)(27,36)(33,40)(35,37);(1,2)(3,5)(4,6)(7,8)(9,11,14,20)(10,16,15,19)(12,21,23,18)(13,22,24,17)(25,28)(26,33)(27,37)(29,34)(30,38)(31,39)(32,40)(35,36);(1,3)(2,5)(4,7)(6,8)(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,19,24,16)(25,27)(26,34)(28,37)(29,33)(30,36)(31,32)(35,38)(39,40);(9,13)(10,18)(11,17)(12,19)(14,24)(15,21)(16,23)(20,22)(25,29)(26,35)(27,33)(28,31)(30,39)(32,37)(34,38)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8424560);