#9789210

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
prime radical	#39722	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4277 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)(11,18,21,16)(12,13,17,22)(19,24,23,20)(25,26,30,28)(27,31,35,33)(29,36,37,32)(34,39,40,38);(1,2,4,6)(3,5,8,7)(10,15)(12,17)(16,18)(19,23)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,2)(3,5)(4,6)(7,8)(26,28)(31,33)(32,36)(38,39);(9,11,14,21)(10,16,15,18)(12,19,17,23)(13,20,22,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,2)(3,7)(4,6)(5,8)(9,12)(10,13)(11,19)(14,17)(15,22)(16,20)(18,24)(21,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,3)(2,5)(4,8)(6,7)(9,13)(10,17)(11,20)(12,15)(14,22)(16,23)(18,19)(21,24)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gh,f^2=h,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, 9696406);