#9232808

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	#50693	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4428 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=gi,e^2=g,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, 9140004);