#8284163

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

4465 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)(11,13,18,16)(12,20,23,17)(19,24,21,22)(25,26)(27,28)(29,37)(30,31)(32,39)(33,36)(34,38)(35,40);(3,7)(5,8)(10,15)(11,18)(17,20)(19,21)(25,27,30,33)(26,28,31,36)(29,34,39,40)(32,35,37,38);(1,2)(3,5)(4,6)(7,8)(10,15)(13,16)(17,20)(22,24)(26,31)(27,33)(32,37)(34,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,18)(10,16,15,13)(12,21,23,19)(17,22,20,24)(25,28)(26,27)(29,38)(30,36)(31,33)(32,40)(34,37)(35,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,23)(10,17,15,20)(11,19,18,21)(13,22,16,24)(25,29)(26,32)(27,34)(28,35)(30,39)(31,37)(33,40)(36,38);(9,13,14,16)(10,18,15,11)(12,22,23,24)(17,21,20,19)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,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,f^c=fgi,e^d=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, 8191359);