#7645737

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

4187 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)(9,10)(11,19)(12,18)(13,15)(14,16)(17,22)(20,23)(21,24)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,16,13,19)(12,20,22,21)(17,23,18,24)(26,27)(31,33)(32,34)(38,39);(9,12)(10,17)(11,20)(13,18)(14,21)(15,22)(16,23)(19,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(9,13,15,10)(11,19,14,16)(12,18,22,17)(20,24,21,23)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,12)(10,18)(11,20)(13,17)(14,21)(15,22)(16,24)(19,23);(1,3,4,7)(2,5,6,8)(9,14,15,11)(10,19,13,16)(12,21,22,20)(17,24,18,23)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=i,d^2=h,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, 7552933);