#9161753

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4453 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=gh,e^2=ghi,f^2=gh,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, 9068949);