#8614182

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

4421 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)(11,16)(12,13)(14,15)(17,22)(18,21)(19,23)(20,24)(25,26,31,32)(27,29,28,33)(30,34,38,39)(35,37,36,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,20)(23,24)(26,32)(27,28)(34,39)(35,36);(2,3)(6,7)(10,15)(12,22)(16,18)(19,24)(25,27,31,28)(26,33,32,29)(30,35,38,36)(34,40,39,37);(4,8)(6,7)(9,11)(10,16)(12,19)(13,23)(14,21)(15,18)(17,20)(22,24)(25,28,31,27)(26,29,32,33)(30,36,38,35)(34,37,39,40);(1,3,5,2)(4,7,8,6)(9,12)(10,17)(11,19)(13,15)(14,22)(16,20)(18,23)(21,24)(25,29)(26,27)(28,32)(30,37)(31,33)(34,35)(36,39)(38,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,17)(10,12,15,22)(11,20,21,23)(16,24,18,19)(25,30,31,38)(26,34,32,39)(27,35,28,36)(29,37,33,40)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8521378);