#9266989

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=gi,e^2=h,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, 9174185);