#8120108

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

4637 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:

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

Polycyclic presentation:

2^9|b^2=i,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8027304);