#8393240

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

4546 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)(10,15)(11,17)(18,22)(19,24)(25,26,30,32)(27,31,34,28)(29,37,40,33)(35,36,38,39);(1,3,4,7)(2,5,6,8)(9,10)(11,12)(13,18)(14,15)(16,17)(19,20)(21,22)(23,24)(25,27)(26,31)(28,32)(29,38)(30,34)(33,36)(35,40)(37,39);(1,2)(3,5)(4,6)(7,8)(13,21)(18,22)(19,24)(20,23)(26,32)(27,34)(33,37)(35,38);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24)(25,28,30,31)(26,27,32,34)(29,39,40,36)(33,38,37,35);(1,2)(3,5)(4,6)(7,8)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,29)(26,33)(27,35)(28,36)(30,40)(31,39)(32,37)(34,38);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,27)(26,31)(28,32)(29,35)(30,34)(33,39)(36,37)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300436);