#9223563

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,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, 9130759);