#7642933

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	#38950	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4414 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,e^2=hi,f^2=g,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, 7550129);