#10188330

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

4449 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,e^2=gh,f^2=hi,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, 10095526);