#10216934

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

4485 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,19,17)(12,15,21,24)(18,22,20,23)(25,26)(27,33)(28,32)(29,34)(30,31)(35,40)(36,37)(38,39);(1,2,4,6)(3,5,8,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,29)(28,36)(30,34)(31,33)(32,40)(35,38)(37,39);(1,2)(3,5)(4,6)(7,8)(26,31)(27,34)(32,39)(35,36);(1,3,4,8)(2,5,6,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,38)(39,40);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,29)(26,27)(28,37)(30,33)(31,34)(32,35)(36,39)(38,40)

Polycyclic presentation:

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

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, 10124130);