#7655323

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4390 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=g,e^2=gh,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, 7562519);