#9717412

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

4405 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=ghi,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, 9624608);