#8618163

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

4549 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=gh,e^2=i,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, 8525359);