#7656907

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

4265 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)(9,10,15,19)(11,20,22,16)(12,14,18,24)(13,17,23,21)(25,26)(27,32)(28,38)(29,33)(30,34)(31,35)(36,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11,15,22)(10,16,19,20)(12,13,18,23)(14,21,24,17)(25,27,31,38)(26,32,35,28)(29,36,30,37)(33,40,34,39);(9,11,15,22)(10,16,19,20)(12,13,18,23)(14,21,24,17)(25,27)(26,28)(29,36)(30,37)(31,38)(32,35)(33,39)(34,40);(1,3,4,7)(2,5,6,8)(9,12,15,18)(10,14,19,24)(11,13,22,23)(16,21,20,17)(25,28)(26,27)(29,39)(30,40)(31,32)(33,36)(34,37)(35,38);(9,13)(10,17)(11,18)(12,22)(14,16)(15,23)(19,21)(20,24)(25,29,31,30)(26,33,35,34)(27,36,38,37)(28,39,32,40);(9,14)(10,18)(11,21)(12,19)(13,20)(15,24)(16,23)(17,22)(25,30,31,29)(26,34,35,33)(27,37,38,36)(28,40,32,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=g,e^2=i,f^2=i,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, 7564103);