#10445005

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

4505 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)(4,6)(5,8)(9,10,15,18)(11,16,12,13)(14,17,22,24)(19,23,20,21)(26,31)(27,32)(29,37)(36,40);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,13,18,16)(14,19,22,20)(17,21,24,23)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,39)(37,38);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,16,18,13)(14,19,22,20)(17,23,24,21)(25,27,30,32)(26,28,31,34)(29,35,37,39)(33,36,38,40);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,16,18,13)(14,19,22,20)(17,21,24,23)(25,26)(27,28)(29,33)(30,31)(32,34)(35,36)(37,38)(39,40);(9,13,15,16)(10,12,18,11)(14,21,22,23)(17,20,24,19)(25,28)(26,32)(27,31)(29,36)(30,34)(33,39)(35,38)(37,40);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,29,30,37)(26,33,31,38)(27,35,32,39)(28,36,34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=ghi,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10352201);