#9780955

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

4359 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)(11,14)(12,19)(13,16)(15,20)(17,23)(18,22)(21,24)(25,26,31,36)(27,37,39,32)(28,33,29,34)(30,35,40,38);(2,5)(6,7)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23)(25,27)(26,32)(28,38)(29,35)(30,34)(31,39)(33,40)(36,37);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,19)(17,21,24,23)(25,28,31,29)(26,33,36,34)(27,38,39,35)(30,32,40,37);(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,39)(38,40);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,29,31,28)(26,34,36,33)(27,38,39,35)(30,32,40,37);(1,3)(2,6)(4,8)(5,7)(9,12,13,20)(10,15,16,19)(11,17,18,24)(14,21,22,23)(25,30,31,40)(26,35,36,38)(27,34,39,33)(28,32,29,37)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,e^2=g,f^2=gh,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, 9688151);