#8726607

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

4562 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gi,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633803);