#9061445

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	#36022	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4464 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8968641);