#8254346

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

4414 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:

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

Polycyclic presentation:

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

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, 8161542);