#8649885

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	#35571	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4496 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8557081);