#8323221

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

4354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8230417);