#10183092

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

4551 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)(3,7)(4,5)(6,8)(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22)(25,26)(27,30)(28,36)(29,32)(31,38)(33,35)(34,40)(37,39);(2,5)(6,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,38)(37,39);(1,2)(3,6)(4,5)(7,8)(10,12)(15,17)(16,20)(22,23)(25,26,29,32)(27,30,35,33)(28,31,38,36)(34,39,37,40);(1,3,4,8)(2,6,5,7)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,38)(37,39);(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,28)(26,31)(27,34)(29,38)(30,39)(32,36)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=hi,e^2=gi,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, 10090288);