#8509515

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

4308 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,e^2=g,f^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, 8416711);