#8630873

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

4444 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)(3,4)(5,8)(6,7)(9,10,16,22)(11,23,24,17)(12,19,13,18)(14,15,21,20)(25,26)(27,28)(29,32)(30,37)(31,33)(34,39)(35,40)(36,38);(1,3)(2,4)(5,6)(7,8)(9,11)(10,17)(12,20)(13,15)(14,18)(16,24)(19,21)(22,23)(25,27,32,31)(26,33,29,28)(30,35,39,38)(34,40,37,36);(1,4)(2,5)(3,7)(6,8)(9,12,16,13)(10,18,22,19)(11,20,24,15)(14,23,21,17)(25,28)(26,31)(27,29)(30,36)(32,33)(34,38)(35,37)(39,40);(1,3,6,5)(2,4,7,8)(9,13,16,12)(10,19,22,18)(11,20,24,15)(14,23,21,17)(25,29,32,26)(27,33,31,28)(30,37,39,34)(35,40,38,36);(1,5,6,3)(2,4,7,8)(9,14,16,21)(10,20,22,15)(11,19,24,18)(12,17,13,23)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40);(1,4)(2,3)(5,7)(6,8)(9,15,16,20)(10,21,22,14)(11,13,24,12)(17,19,23,18)(25,31,32,27)(26,33,29,28)(30,38,39,35)(34,40,37,36)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8538069);