#8800215

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

4641 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8707411);