#8260735

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	#36410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4432 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:

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

Polycyclic presentation:

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

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, 8167931);