#8933865

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	#60048	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4446 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841061);