#8436216

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

4436 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,d^2=g,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, 8343412);