#7656785

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	#35736	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4335 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)(9,10)(11,22)(12,23)(13,19)(14,18)(15,20)(16,21)(17,24)(28,36)(31,38)(35,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,13)(14,20)(15,23)(16,24)(18,19)(21,22)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(9,12,16,14)(10,18,21,23)(11,13,24,20)(15,17,19,22)(25,27,30,32)(26,29,33,34)(28,35,36,39)(31,37,38,40);(1,3,4,7)(2,6,5,8)(9,13,16,20)(10,19,21,15)(11,12,24,14)(17,18,22,23)(25,28)(26,31)(27,35)(29,37)(30,36)(32,39)(33,38)(34,40);(9,14,16,12)(10,18,21,23)(11,20,24,13)(15,17,19,22)(25,26)(27,29)(28,31)(30,33)(32,34)(35,37)(36,38)(39,40);(9,15,16,19)(10,20,21,13)(11,23,24,18)(12,17,14,22)(25,29)(26,32)(27,33)(28,37)(30,34)(31,39)(35,38)(36,40)

Polycyclic presentation:

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

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, 7563981);