#9127650

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	#38900	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4481 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9034846);