#8720692

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

4586 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)(9,10,14,18)(11,13,17,15)(12,20,21,16)(19,22,24,23)(25,26)(27,34)(28,37)(29,39)(30,32)(31,33)(35,40)(36,38);(1,2,5,4)(3,7,8,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,27,31,30)(26,32,33,34)(28,38,39,35)(29,36,37,40);(1,3,5,8)(2,6,4,7)(12,21)(16,20)(19,24)(22,23)(26,33)(28,39)(32,34)(35,38);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,19)(13,23)(14,21)(15,22)(17,24)(18,20)(25,28)(26,29)(27,35)(30,38)(31,39)(32,40)(33,37)(34,36);(9,11)(10,13)(12,19)(14,17)(15,18)(16,23)(20,22)(21,24)(25,29)(26,28)(27,36)(30,40)(31,37)(32,38)(33,39)(34,35);(9,13)(10,17)(11,18)(12,22)(14,15)(16,19)(20,24)(21,23)(25,30,31,27)(26,32,33,34)(28,35,39,38)(29,36,37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,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, 8627888);