#9063244

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	#50677	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,e^2=ghi,f^2=h,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, 8970440);