#10166217

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

4636 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:

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

Polycyclic presentation:

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

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, 10073413);