#9789198

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

4330 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)(28,35)(29,32)(30,31)(34,40)(36,37)(38,39);(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)(26,28)(29,34)(30,35)(31,33)(32,36)(37,38)(39,40);(1,2)(3,5)(4,6)(7,8)(26,31)(28,33)(32,39)(36,40);(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,39)(38,40);(1,2)(3,7)(4,6)(5,8)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,29,30,38)(26,32,31,39)(27,34,35,37)(28,36,33,40);(1,3)(2,5)(4,8)(6,7)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,27)(26,28)(29,37)(30,35)(31,33)(32,40)(34,38)(36,39)

Polycyclic presentation:

2^9|b^2=ghi,d^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, 9696394);