#8429275

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

4166 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=gh,e^2=g,f^2=gi,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, 8336471);