#7653138

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4492 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,e^2=gi,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, 7560334);