#8425315

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

4495 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=g,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, 8332511);