#7653222

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4318 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)(9,10,15,19)(11,17,12,16)(13,14,18,24)(20,23,22,21)(25,26,32,30)(27,35,31,33)(28,29,34,39)(36,37,38,40);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,16,19,17)(13,20,18,22)(14,21,24,23)(25,27,32,31)(26,33,30,35)(28,38,34,36)(29,37,39,40);(9,11,15,12)(10,16,19,17)(13,20,18,22)(14,21,24,23)(25,28)(26,29)(27,36)(30,39)(31,38)(32,34)(33,40)(35,37);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,17,19,16)(13,22,18,20)(14,23,24,21)(25,29,32,39)(26,34,30,28)(27,37,31,40)(33,36,35,38);(9,13)(10,14)(11,20)(12,22)(15,18)(16,21)(17,23)(19,24)(25,30,32,26)(27,35,31,33)(28,39,34,29)(36,37,38,40);(9,14)(10,18)(11,21)(12,23)(13,19)(15,24)(16,22)(17,20)(25,31,32,27)(26,33,30,35)(28,38,34,36)(29,40,39,37)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=i,f^2=i,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, 7560418);