#9222207

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	#43257	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4392 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=hi,e^2=g,f^2=gh,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, 9129403);