#8720631

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

4509 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=i,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627827);