#8702610

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	#43247	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8609806);