#9224228

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4367 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=ghi,e^2=h,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, 9131424);