#9271811

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

4525 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=h,e^2=hi,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, 9179007);