#8459126

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

4634 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8366322);