#9105309

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	#35932	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4319 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)(11,17,18,13)(12,16,20,23)(19,24,21,22)(25,26,30,34)(27,32,37,31)(28,38,33,29)(35,36,39,40);(1,2,5,3)(4,6,8,7)(10,15)(11,18)(16,23)(19,21)(25,26)(27,31)(28,38)(29,33)(30,34)(32,37)(35,40)(36,39);(2,3)(6,7)(12,20)(16,23)(19,21)(22,24)(25,27,30,37)(26,31,34,32)(28,35,33,39)(29,36,38,40);(2,3)(4,8)(9,11,14,18)(10,13,15,17)(12,21,20,19)(16,24,23,22)(25,28)(26,29)(27,35)(30,33)(31,36)(32,40)(34,38)(37,39);(1,3,5,2)(4,7,8,6)(9,12,14,20)(10,16,15,23)(11,19,18,21)(13,22,17,24)(25,27)(26,32)(28,39)(29,36)(30,37)(31,34)(33,35)(38,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,22)(14,17)(15,18)(16,19)(20,24)(21,23)(25,29)(26,33)(27,36)(28,34)(30,38)(31,39)(32,35)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=gh,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=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, 9012505);