#7653164

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4320 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=gi,f^2=gi,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, 7560360);