#8254554

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4145 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=h,e^2=hi,f^2=gi,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, 8161750);