#7657472

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=i,e^2=ghi,f^2=hi,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, 7564668);