#8689794

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4463 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8596990);