#9062555

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	#44181	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4356 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)(11,19)(12,21)(13,17)(14,23)(15,16)(18,24)(20,22)(25,26,30,29)(27,32,35,31)(28,33,38,37)(34,40,36,39);(1,2)(3,7)(4,5)(6,8)(9,11,15,21)(10,12,16,19)(13,14,20,24)(17,18,22,23)(25,27,30,35)(26,31,29,32)(28,36,38,34)(33,40,37,39);(2,5)(3,8)(10,16)(12,19)(13,20)(14,24)(25,27,30,35)(26,32,29,31)(28,36,38,34)(33,39,37,40);(1,3,4,8)(2,6,5,7)(9,12)(10,11)(13,23)(14,22)(15,19)(16,21)(17,24)(18,20)(25,27)(26,31)(28,36)(29,32)(30,35)(33,40)(34,38)(37,39);(1,3)(2,6)(4,8)(5,7)(9,13,15,20)(10,17,16,22)(11,14,21,24)(12,18,19,23)(25,28,30,38)(26,33,29,37)(27,34,35,36)(31,39,32,40);(1,2)(3,6)(4,5)(7,8)(9,14)(10,18)(11,20)(12,22)(13,21)(15,24)(16,23)(17,19)(25,29,30,26)(27,32,35,31)(28,37,38,33)(34,40,36,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969751);