#7643453

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4516 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550649);