#9682734

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	#57279	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4163 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,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,d^c=dh,f^e=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, 9589930);