#8533529

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

4400 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440725);