#8107553

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

4485 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

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, 8014749);