#8425610

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4196 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8332806);