#7657284

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

4366 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)(12,20)(17,22)(19,21)(23,24)(25,26,32,36)(27,37,30,33)(28,39,38,34)(29,31,35,40);(1,2)(3,6)(4,7)(5,8)(9,10,15,14)(11,18,13,16)(12,17,20,22)(19,24,21,23)(25,27,32,30)(26,33,36,37)(28,29,38,35)(31,39,40,34);(1,3,5,4)(2,6,8,7)(9,11,15,13)(10,16,14,18)(12,19,20,21)(17,23,22,24)(25,28)(26,34)(27,29)(30,35)(31,33)(32,38)(36,39)(37,40);(1,3,5,4)(2,6,8,7)(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,29,32,35)(26,31,36,40)(27,38,30,28)(33,39,37,34);(1,4,5,3)(2,7,8,6)(9,13,15,11)(10,16,14,18)(12,21,20,19)(17,23,22,24)(25,30,32,27)(26,33,36,37)(28,35,38,29)(31,39,40,34);(9,14,15,10)(11,18,13,16)(12,22,20,17)(19,24,21,23)(25,31)(26,35)(27,39)(28,33)(29,36)(30,34)(32,40)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=g,e^2=gi,f^2=i,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, 7564480);