#8716820

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

4592 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)(11,19)(12,14)(13,16)(15,17)(18,23)(20,21)(22,24)(25,26,31,30)(27,35,29,32)(28,33,38,34)(36,40,37,39);(1,2,5,8)(3,4,7,6)(9,11,15,13)(10,16,17,19)(12,22,23,20)(14,21,18,24)(25,27,31,29)(26,32,30,35)(28,36,38,37)(33,39,34,40);(1,3)(2,6)(4,8)(5,7)(10,17)(12,23)(16,19)(20,22)(25,28,31,38)(26,33,30,34)(27,36,29,37)(32,39,35,40);(1,4,5,6)(2,7,8,3)(9,12,15,23)(10,18,17,14)(11,20,13,22)(16,21,19,24)(25,28)(26,34)(27,36)(29,37)(30,33)(31,38)(32,40)(35,39);(1,3)(2,4)(5,7)(6,8)(9,13,15,11)(10,19,17,16)(12,22,23,20)(14,21,18,24)(25,29,31,27)(26,32,30,35)(28,37,38,36)(33,39,34,40);(1,4)(2,7)(3,8)(5,6)(9,14,15,18)(10,12,17,23)(11,21,13,24)(16,22,19,20)(25,30,31,26)(27,35,29,32)(28,34,38,33)(36,40,37,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=hi,e^2=g,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, 8624016);