#8612909

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

4337 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8520105);