#8253895

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

4514 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8161091);