#9250227

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	#53782	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4069 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gi,d^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,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9157423);