#8436967

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4361 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=g,e^2=gh,f^2=g,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, 8344163);