#8722229

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×D₄²	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4499 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8629425);