#8284463

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

4783 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8191659);