#7641958

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	C₂⁵	C₂⁴
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)(3,7,8,5)(10,13)(11,17)(16,21)(18,23)(28,35)(32,37)(34,39)(38,40);(1,3,4,8)(2,5,6,7)(12,20)(16,21)(18,23)(22,24)(26,30)(27,33)(32,37)(34,39);(9,10,14,13)(11,15,17,19)(12,16,20,21)(18,22,23,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24);(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(9,13,14,10)(11,19,17,15)(12,21,20,16)(18,24,23,22)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=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, 7549154);