#8289366

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	#59157	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 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)(11,13)(12,20)(14,16)(15,19)(17,23)(18,24)(21,22)(25,26,31,27)(28,35,38,32)(29,33,30,34)(36,40,37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,21)(13,16)(14,19)(17,22)(18,23)(20,24)(26,27)(32,35)(33,34)(39,40);(2,5)(6,8)(10,16)(13,15)(17,20)(22,24)(25,27,31,26)(28,35,38,32)(29,34,30,33)(36,40,37,39);(1,3,4,7)(2,6,5,8)(9,12,14,23)(10,17,16,20)(11,18,19,21)(13,22,15,24)(25,28,31,38)(26,32,27,35)(29,36,30,37)(33,39,34,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,21)(13,16)(14,19)(17,22)(18,23)(20,24)(25,29,31,30)(26,33,27,34)(28,36,38,37)(32,39,35,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,15)(10,11,16,19)(12,22,23,24)(17,18,20,21)(25,30,31,29)(26,34,27,33)(28,37,38,36)(32,40,35,39)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8196562);