#9780521

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

4671 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9687717);