#7644321

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4084 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7551517);