#9333327

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

4767 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)(9,10,14,13)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,26,31,28)(27,32,34,29)(30,33,39,36)(35,40,38,37);(1,2,5,8)(3,6,7,4)(10,13)(15,17)(16,21)(22,23)(25,27,31,34)(26,32,28,29)(30,38,39,35)(33,37,36,40);(1,2,5,8)(3,4,7,6)(12,20)(16,21)(18,24)(22,23)(27,34)(29,32)(35,38)(37,40);(1,3,5,7)(2,6,8,4)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,24)(21,23)(25,28,31,26)(27,29,34,32)(30,36,39,33)(35,37,38,40);(1,2)(3,6)(4,7)(5,8)(9,12,14,20)(10,16,13,21)(11,18,19,24)(15,22,17,23)(25,29)(26,27)(28,34)(30,37)(31,32)(33,35)(36,38)(39,40);(1,4,5,6)(2,7,8,3)(9,13,14,10)(11,17,19,15)(12,21,20,16)(18,23,24,22)(25,30,31,39)(26,33,28,36)(27,35,34,38)(29,37,32,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9240523);