#8664917

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

4568 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8572113);