#7642566

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	#38087	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)(10,13)(11,17)(16,20)(18,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(1,2,4,5)(3,6,7,8)(9,10,14,13)(11,15,17,19)(12,16,21,20)(18,22,23,24)(26,30)(31,33)(32,36)(38,39);(9,10,14,13)(11,15,17,19)(12,20,21,16)(18,24,23,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23);(9,12,14,21)(10,16,13,20)(11,18,17,23)(15,22,19,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,19,17,15)(12,20,21,16)(18,24,23,22)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,e^2=h,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, 7549762);