#8517422

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

4576 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)(3,5,7,8)(9,10,14,16)(11,15,18,12)(13,17,22,23)(19,21,24,20)(26,31)(27,33)(28,35)(37,40);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,21)(20,23)(22,24)(25,26,30,31)(27,29,33,38)(28,32,35,39)(34,37,36,40);(1,2)(3,5)(4,6)(7,8)(10,16)(11,18)(17,23)(19,24)(25,27,30,33)(26,29,31,38)(28,36,35,34)(32,40,39,37);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,21)(14,15)(16,18)(17,24)(19,23)(20,22)(25,26)(27,29)(28,32)(30,31)(33,38)(34,37)(35,39)(36,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,23)(22,24)(25,28,30,35)(26,32,31,39)(27,34,33,36)(29,37,38,40);(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,29)(26,33)(27,31)(28,37)(30,38)(32,36)(34,39)(35,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,e^2=g,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, 8424618);