#8186935

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

4444 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=i,e^2=gh,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, 8094131);