#8703948

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4254 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)(11,14)(12,16)(13,17)(15,19)(18,23)(20,22)(21,24)(25,26,31,35)(27,29,36,32)(28,37,40,33)(30,34,38,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,24)(21,22)(25,27,31,36)(26,32,35,29)(28,38,40,30)(33,39,37,34);(1,3)(2,5)(4,8)(6,7)(9,10,13,17)(11,15,19,14)(12,16,22,20)(18,24,21,23)(25,27,31,36)(26,32,35,29)(28,38,40,30)(33,39,37,34);(3,8)(5,6)(9,10)(11,15)(12,20)(13,17)(14,19)(16,22)(18,23)(21,24)(25,28,31,40)(26,33,35,37)(27,30,36,38)(29,39,32,34);(1,2)(3,5)(4,7)(6,8)(9,11)(10,15)(12,21)(13,19)(14,17)(16,23)(18,22)(20,24)(25,29)(26,27)(28,39)(30,33)(31,32)(34,40)(35,36)(37,38);(1,3)(2,6)(4,8)(5,7)(9,12,13,22)(10,16,17,20)(11,18,19,21)(14,23,15,24)(25,30)(26,34)(27,28)(29,37)(31,38)(32,33)(35,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,f^2=gh,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, 8611144);