#8678987

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

4189 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)(3,5)(4,7)(6,8)(10,15)(11,18)(17,24)(19,21)(25,26,30,29)(27,31,33,35)(28,32,36,37)(34,38,39,40);(2,3)(6,7)(9,10,14,15)(11,16,18,12)(13,17,23,24)(19,22,21,20)(27,33)(31,35)(34,39)(38,40);(1,2,5,3)(4,6,8,7)(25,27,30,33)(26,31,29,35)(28,34,36,39)(32,38,37,40);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,15)(13,21)(14,18)(17,20)(19,23)(22,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,4,5,8)(2,6,3,7)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,21,24,19)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(9,13,14,23)(10,17,15,24)(11,19,18,21)(12,20,16,22)(25,29,30,26)(27,35,33,31)(28,32,36,37)(34,38,39,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=h,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,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, 8586183);