#9229588

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4339 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)(11,18)(12,17)(13,22)(14,15)(16,21)(19,20)(23,24)(25,26,31,29)(27,35,37,32)(28,38,36,33)(30,34,40,39);(1,2)(3,4)(5,7)(6,8)(10,15)(12,22)(16,18)(19,23)(25,27,31,37)(26,32,29,35)(28,30,36,40)(33,34,38,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,18)(17,24)(22,23)(25,26,31,29)(27,32,37,35)(28,33,36,38)(30,34,40,39);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,23)(13,24)(14,21)(15,18)(17,20)(19,22)(25,28)(26,33)(27,30)(29,38)(31,36)(32,34)(35,39)(37,40);(1,4,5,6)(2,3,7,8)(9,12,14,22)(10,17,15,13)(11,19,21,23)(16,24,18,20)(25,29,31,26)(27,35,37,32)(28,33,36,38)(30,34,40,39);(1,3)(2,4)(5,8)(6,7)(9,13,14,17)(10,12,15,22)(11,20,21,24)(16,19,18,23)(25,30)(26,34)(27,36)(28,37)(29,39)(31,40)(32,38)(33,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,e^2=gi,f^2=h,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, 9136784);