#8935970

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

4403 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8843166);