#9123783

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	#49750	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4537 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9030979);