#7642644

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

4368 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)(11,13)(12,16)(14,15)(17,22)(18,20)(19,24)(21,23)(25,26,30,34)(27,33,29,31)(28,32,38,36)(35,40,37,39);(1,2,5,6)(3,7,4,8)(10,15)(11,17)(16,23)(18,19);(1,3,5,4)(2,7,6,8)(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,26)(27,31)(28,36)(29,33)(30,34)(32,38)(35,40)(37,39);(25,27,30,29)(26,31,34,33)(28,35,38,37)(32,39,36,40);(1,4,5,3)(2,8,6,7)(9,12,14,21)(10,16,15,23)(11,18,17,19)(13,20,22,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,29,30,27)(26,33,34,31)(28,37,38,35)(32,40,36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=h,e^2=g,f^2=h,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, 7549840);