#8720793

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

4630 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=gi,e^2=g,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, 8627989);