#10147062

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

4517 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,e^2=gi,f^2=g,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, 10054258);