#8506034

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	#39169	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4477 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=gh,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8413230);