#8657986

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	#48564	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4333 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8565182);