#7644253

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

4077 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)(9,10,15,18)(11,16,14,19)(12,17,23,13)(20,21,22,24)(26,28)(27,33)(32,36)(35,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,19)(12,13)(14,16)(15,18)(17,23)(20,21)(22,24)(25,26,30,28)(27,34,33,31)(29,32,37,36)(35,40,39,38);(9,11,15,14)(10,16,18,19)(12,20,23,22)(13,24,17,21)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(9,12)(10,17)(11,20)(13,18)(14,22)(15,23)(16,21)(19,24)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(9,13,15,17)(10,12,18,23)(11,21,14,24)(16,22,19,20);(1,3,4,7)(2,5,6,8)(9,14,15,11)(10,16,18,19)(12,22,23,20)(13,24,17,21)(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=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7551449);