#9232843

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

4269 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=gi,e^2=gh,f^2=gi,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, 9140039);