#8716764

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

4612 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)(11,17)(12,18)(13,16)(14,15)(19,22)(20,21)(23,24)(25,26,31,35)(27,30,34,32)(28,33,38,29)(36,37,40,39);(1,2,5,8)(3,4,7,6)(10,15)(11,18)(16,24)(19,21)(25,27,31,34)(26,32,35,30)(28,36,38,40)(29,37,33,39);(1,3)(2,6)(4,8)(5,7)(10,15)(11,18)(16,24)(19,21)(25,28,31,38)(26,33,35,29)(27,36,34,40)(30,37,32,39);(1,4,5,6)(2,7,8,3)(9,11)(10,12)(13,21)(14,18)(15,17)(16,22)(19,23)(20,24)(25,28)(26,29)(27,36)(30,39)(31,38)(32,37)(33,35)(34,40);(1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(13,22)(14,17)(15,18)(16,21)(19,24)(20,23)(25,29)(26,28)(27,37)(30,40)(31,33)(32,36)(34,39)(35,38);(1,4)(2,7)(3,8)(5,6)(9,13,14,23)(10,16,15,24)(11,19,18,21)(12,20,17,22)(25,30,31,32)(26,34,35,27)(28,37,38,39)(29,36,33,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=i,f^2=g,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, 8623960);