#9057877

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

4628 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8965073);