#7646854

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4218 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7554050);