#8184788

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

4575 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,26,32,29)(27,33,30,39)(28,37,36,34)(31,35,38,40);(2,6)(3,8)(10,11)(15,17)(16,18)(22,23)(25,27,32,30)(26,33,29,39)(28,31,36,38)(34,40,37,35);(1,3)(2,4)(5,8)(6,7)(13,21)(16,18)(19,24)(22,23)(25,28)(26,34)(27,31)(29,37)(30,38)(32,36)(33,40)(35,39);(1,3)(2,4)(5,8)(6,7)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,29,32,26)(27,33,30,39)(28,37,36,34)(31,40,38,35);(1,4)(2,3)(5,7)(6,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,30,32,27)(26,33,29,39)(28,38,36,31)(34,40,37,35);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,31,32,38)(26,35,29,40)(27,36,30,28)(33,34,39,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8091984);