#7654232

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

4477 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)(11,17)(12,22)(13,15)(14,18)(16,20)(19,21)(23,24)(25,26,31,35)(27,36,38,32)(28,29,33,39)(30,34,40,37);(1,2,5,6)(3,7,4,8)(9,11,16,23)(10,17,20,24)(12,21,14,13)(15,22,19,18)(25,27,31,38)(26,32,35,36)(28,30,33,40)(29,37,39,34);(1,3,5,4)(2,7,6,8)(9,12,16,14)(10,18,20,22)(11,21,23,13)(15,24,19,17)(25,26,31,35)(27,36,38,32)(28,29,33,39)(30,34,40,37);(1,4,5,3)(2,8,6,7)(9,13)(10,19)(11,12)(14,23)(15,20)(16,21)(17,18)(22,24)(25,28,31,33)(26,29,35,39)(27,30,38,40)(32,37,36,34);(9,14,16,12)(10,18,20,22)(11,13,23,21)(15,24,19,17)(25,29)(26,33)(27,37)(28,35)(30,36)(31,39)(32,40)(34,38);(9,15,16,19)(10,13,20,21)(11,22,23,18)(12,24,14,17)(25,30)(26,34)(27,33)(28,38)(29,32)(31,40)(35,37)(36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561428);