#7658865

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

4474 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)(11,18)(12,17)(13,19)(14,15)(16,22)(20,21)(23,24)(25,26,31,32)(27,33,37,40)(28,30,36,34)(29,35,38,39);(1,2)(3,6)(4,7)(5,8)(10,15)(12,19)(16,18)(20,24)(26,32)(27,37)(28,36)(35,39);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,20)(13,21)(14,22)(15,18)(17,23)(19,24)(25,27)(26,33)(28,38)(29,36)(30,39)(31,37)(32,40)(34,35);(1,3,5,4)(2,6,8,7)(9,12,14,19)(10,17,15,13)(11,20,22,24)(16,23,18,21)(25,28,31,36)(26,34,32,30)(27,38,37,29)(33,35,40,39);(1,3,5,4)(2,6,8,7)(9,11)(10,18)(12,20)(13,23)(14,22)(15,16)(17,21)(19,24)(25,29,31,38)(26,35,32,39)(27,36,37,28)(30,33,34,40);(1,4,5,3)(2,7,8,6)(9,13,14,17)(10,19,15,12)(11,21,22,23)(16,24,18,20)(25,30,31,34)(26,36,32,28)(27,39,37,35)(29,40,38,33)

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7566061);