#8703486

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4470 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=gh,e^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, 8610682);