#7653409

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

4290 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)(11,17)(18,23)(19,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,39)(38,40);(1,2,5,6)(3,7,4,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23)(25,27,31,39)(26,32,36,37)(28,35,34,38)(29,30,33,40);(1,3,5,4)(2,7,6,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23)(25,28,31,34)(26,33,36,29)(27,38,39,35)(30,37,40,32);(1,3,5,4)(2,7,6,8)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,21,23,20)(25,29,31,33)(26,34,36,28)(27,30,39,40)(32,38,37,35);(1,3,5,4)(2,7,6,8)(9,12,14,16)(10,17,15,11)(13,21,22,20)(18,19,23,24)(25,26)(27,32)(28,33)(29,34)(30,38)(31,36)(35,40)(37,39);(1,4,5,3)(2,8,6,7)(9,13,14,22)(10,18,15,23)(11,19,17,24)(12,20,16,21)(25,30)(26,35)(27,33)(28,37)(29,39)(31,40)(32,34)(36,38)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=g,e^2=gi,f^2=gi,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, 7560605);