#7656900

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

4615 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=g,e^2=h,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, 7564096);