#7642557

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4432 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)(9,10,15,13)(11,18,14,16)(12,20,22,17)(19,23,21,24)(28,36)(31,38)(34,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,16,13,18)(12,19,22,21)(17,23,20,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(10,13)(16,18)(17,20)(23,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(9,12)(10,17)(11,19)(13,20)(14,21)(15,22)(16,23)(18,24);(9,13,15,10)(11,18,14,16)(12,20,22,17)(19,24,21,23)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,3,4,7)(2,5,6,8)(9,14,15,11)(10,18,13,16)(12,21,22,19)(17,24,20,23)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

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, 7549753);