#8422105

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

4460 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8329301);