#7654242

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4371 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)(11,20)(12,13)(14,18)(15,17)(16,19)(21,22)(23,24)(25,26,31,29)(27,34,32,28)(30,33,38,37)(35,40,39,36);(1,3,5,8)(2,6,4,7)(9,11,16,15)(10,17,19,20)(12,23,14,21)(13,22,18,24)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,40)(38,39);(1,2,5,4)(3,6,8,7)(9,12,16,14)(10,13,19,18)(11,21,15,23)(17,24,20,22)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,40)(38,39);(1,2,5,4)(3,6,8,7)(9,13,16,18)(10,12,19,14)(11,22,15,24)(17,23,20,21)(25,28,31,34)(26,32,29,27)(30,36,38,40)(33,39,37,35);(1,2,5,4)(3,6,8,7)(9,14,16,12)(10,18,19,13)(11,21,15,23)(17,24,20,22)(25,29,31,26)(27,34,32,28)(30,33,38,37)(35,36,39,40);(1,4,5,2)(3,7,8,6)(9,15,16,11)(10,17,19,20)(12,23,14,21)(13,24,18,22)(25,30,31,38)(26,33,29,37)(27,35,32,39)(28,36,34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=g,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, 7561438);