#8581816

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

4353 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8489012);