#8184869

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

4500 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)(3,5)(4,6)(7,8)(9,10,16,15)(11,13,17,12)(14,22,23,18)(19,20,24,21)(26,28)(27,33)(32,36)(35,39);(2,3)(6,7)(9,11)(10,13)(12,15)(14,19)(16,17)(18,21)(20,22)(23,24)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(4,8)(6,7)(9,12,16,13)(10,17,15,11)(14,20,23,21)(18,24,22,19)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38);(1,3,5,2)(4,7,8,6)(9,13,16,12)(10,17,15,11)(14,21,23,20)(18,24,22,19)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(9,14,16,23)(10,18,15,22)(11,19,17,24)(12,20,13,21)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(1,4)(2,6)(3,7)(5,8)(9,15,16,10)(11,12,17,13)(14,22,23,18)(19,20,24,21)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

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

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, 8092065);