#9712433

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4385 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9619629);