#8762057

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

4936 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,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, 8669253);