#9224233

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	#36316	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4575 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)(10,14)(15,17)(16,20)(23,24)(25,26)(27,33)(28,36)(29,34)(30,31)(32,38)(35,37)(39,40);(3,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(26,31)(27,34)(32,36)(35,40);(1,2)(3,5)(4,6)(7,8)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(9,10)(11,17)(12,20)(13,14)(15,19)(16,22)(18,23)(21,24)(25,28,30,38)(26,32,31,36)(27,35,34,40)(29,37,33,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,22)(20,23)(25,26)(27,29)(28,32)(30,31)(33,34)(35,37)(36,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,22)(14,20)(15,23)(17,24)(19,21)(25,29)(26,27)(28,37)(30,33)(31,34)(32,35)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=hi,d^2=i,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, 9131429);