#9224374

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	#49984	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4492 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)(3,8)(4,7)(5,6)(9,10,15,17)(11,16,20,24)(12,21,13,14)(18,19,22,23)(26,31)(27,32)(29,37)(35,39);(1,3,5,7)(2,4,6,8)(9,11,15,20)(10,16,17,24)(12,22,13,18)(14,19,21,23)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(9,12,15,13)(10,14,17,21)(11,18,20,22)(16,23,24,19)(25,27,30,32)(26,28,31,36)(29,34,37,40)(33,35,38,39);(2,6)(4,8)(9,13,15,12)(10,14,17,21)(11,18,20,22)(16,19,24,23)(25,26)(27,28)(29,38)(30,31)(32,36)(33,37)(34,39)(35,40);(1,2)(3,4)(5,6)(7,8)(9,11)(10,16)(12,18)(13,22)(14,23)(15,20)(17,24)(19,21)(25,28)(26,32)(27,31)(29,39)(30,36)(33,34)(35,37)(38,40);(1,4)(2,7)(3,6)(5,8)(9,14)(10,13)(11,19)(12,17)(15,21)(16,18)(20,23)(22,24)(25,29)(26,33)(27,34)(28,35)(30,37)(31,38)(32,40)(36,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,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, 9131570);