#8157672

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

4519 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8064868);