#7647105

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=i,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7554301);