#8456818

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

4609 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,d^2=h,e^2=hi,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, 8364014);