#8517334

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

4677 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)(9,10,14,15)(11,12,19,17)(13,16,22,23)(18,20,21,24)(26,29)(27,28)(34,37)(35,36);(3,7)(5,8)(9,10,14,15)(11,17,19,12)(13,16,22,23)(18,24,21,20)(25,26,31,29)(27,33,28,32)(30,34,38,37)(35,40,36,39);(1,2)(3,5)(4,6)(7,8)(10,15)(12,17)(16,23)(20,24)(25,27,31,28)(26,32,29,33)(30,35,38,36)(34,39,37,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,12)(13,21)(14,19)(15,17)(16,24)(18,22)(20,23)(25,28,31,27)(26,33,29,32)(30,36,38,35)(34,40,37,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,29,31,26)(27,33,28,32)(30,37,38,34)(35,40,36,39);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,30,31,38)(26,34,29,37)(27,35,28,36)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=g,e^2=g,f^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, 8424530);