#9040974

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	#35811	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4392 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)(11,15)(12,17)(13,18)(14,16)(19,20)(21,22)(23,24)(25,26,31,29)(27,30,35,32)(28,36,39,33)(34,40,37,38);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,23)(13,22)(14,20)(16,19)(17,24)(18,21)(25,27,31,35)(26,32,29,30)(28,37,39,34)(33,38,36,40);(2,5)(3,7)(10,16)(11,20)(13,17)(21,23)(26,29)(30,32)(33,36)(38,40);(2,5)(6,8)(9,12)(10,17)(11,21)(13,16)(14,18)(15,22)(19,24)(20,23)(25,28,31,39)(26,33,29,36)(27,34,35,37)(30,38,32,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,17)(10,18,16,12)(11,22,20,24)(15,21,19,23)(25,29,31,26)(27,30,35,32)(28,33,39,36)(34,40,37,38);(1,3)(2,6)(4,7)(5,8)(9,11)(10,19)(12,21)(13,22)(14,20)(15,16)(17,24)(18,23)(25,30)(26,27)(28,38)(29,35)(31,32)(33,34)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 8948170);