#8434461

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

4537 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^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, 8341657);