#9028415

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

4487 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=gh,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8935611);