#9434714

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	#54153	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4061 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=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, 9341910);