#8270597

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

4411 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=i,e^2=ghi,f^2=gi,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, 8177793);