#8466933

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	#36369	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4385 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)(3,6)(4,7)(5,8)(10,14)(11,17)(16,22)(18,19)(25,26,32,30)(27,37,39,33)(28,35,29,34)(31,36,38,40);(1,3,5,7)(2,6,8,4)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,27)(26,33)(28,38)(29,31)(30,37)(32,39)(34,40)(35,36);(1,3)(2,4)(5,7)(6,8)(25,28,32,29)(26,34,30,35)(27,38,39,31)(33,40,37,36);(1,3)(2,6)(4,8)(5,7)(9,11,13,17)(10,15,14,21)(12,19,20,18)(16,24,22,23)(25,29,32,28)(26,35,30,34)(27,38,39,31)(33,40,37,36);(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,30,32,26)(27,37,39,33)(28,35,29,34)(31,40,38,36);(1,4,5,6)(2,7,8,3)(9,12,13,20)(10,16,14,22)(11,18,17,19)(15,23,21,24)(25,31,32,38)(26,36,30,40)(27,29,39,28)(33,35,37,34)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=g,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8374129);