#9789182

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

4538 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:

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

Polycyclic presentation:

2^9|b^2=ghi,d^2=gh,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, 9696378);