#8712240

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×#546	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4497 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=h,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, 8619436);