#7645809

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	#35144	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4308 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)(11,12)(18,22)(19,20)(26,30)(27,33)(32,37)(34,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(9,11,14,12)(10,16,15,17)(13,19,21,20)(18,23,22,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,17,15,16)(13,19,21,20)(18,23,22,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(9,13,14,21)(10,18,15,22)(11,19,12,20)(16,23,17,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)

Polycyclic presentation:

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

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, 7553005);