#10340276

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

4744 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10247472);