#7647232

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

4514 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:

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

Polycyclic presentation:

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

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, 7554428);