#10444988

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

4446 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)(3,7)(4,6)(5,8)(9,10,15,18)(11,16,12,13)(14,17,22,24)(19,23,20,21)(26,31)(28,32)(29,37)(35,38);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,13,18,16)(14,19,22,20)(17,21,24,23)(26,31)(28,32)(34,39)(36,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,16,18,13)(14,19,22,20)(17,23,24,21)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,38)(37,39);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,16,18,13)(14,19,22,20)(17,21,24,23)(25,27,30,33)(26,32,31,28)(29,38,37,35)(34,36,39,40);(9,13,15,16)(10,12,18,11)(14,21,22,23)(17,20,24,19)(25,28,30,32)(26,33,31,27)(29,36,37,40)(34,38,39,35);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,29,30,37)(26,34,31,39)(27,35,33,38)(28,36,32,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,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, 10352184);