#9726465

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

4482 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)(3,4)(5,8)(6,7)(9,10,15,14)(11,19,22,16)(12,17,21,23)(13,18,20,24)(25,26,31,37)(27,36,30,32)(28,34,39,33)(29,35,40,38);(1,3,5,7)(2,4,8,6)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,18)(23,24)(25,27,31,30)(26,32,37,36)(28,35,39,38)(29,33,40,34);(1,3)(2,6)(4,8)(5,7)(25,28,31,39)(26,33,37,34)(27,35,30,38)(29,32,40,36);(1,2,5,8)(3,4,7,6)(9,12,15,21)(10,17,14,23)(11,20,22,13)(16,18,19,24)(25,28)(26,34)(27,35)(29,32)(30,38)(31,39)(33,37)(36,40);(1,3)(2,4)(5,7)(6,8)(9,13,15,20)(10,18,14,24)(11,21,22,12)(16,17,19,23)(25,29)(26,35)(27,34)(28,32)(30,33)(31,40)(36,39)(37,38);(1,4)(2,7)(3,8)(5,6)(9,14,15,10)(11,19,22,16)(12,23,21,17)(13,18,20,24)(25,30,31,27)(26,36,37,32)(28,38,39,35)(29,34,40,33)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=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, 9633661);