#7712413

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

4356 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)(10,16)(14,18)(17,20)(22,24)(25,26,31,35)(27,36,32,28)(29,33,34,30)(37,38,40,39);(1,2,5,6)(3,7,4,8)(9,10)(11,20)(12,17)(13,18)(14,19)(15,16)(21,24)(22,23)(25,27,31,32)(26,28,35,36)(29,37,34,40)(30,38,33,39);(1,3,5,4)(2,7,6,8)(9,11,15,12)(10,17,16,20)(13,23,19,21)(14,22,18,24)(25,28,31,36)(26,32,35,27)(29,38,34,39)(30,40,33,37);(1,3,5,4)(2,7,6,8)(9,12,15,11)(10,17,16,20)(13,23,19,21)(14,24,18,22)(25,29,31,34)(26,33,35,30)(27,37,32,40)(28,39,36,38);(1,4,5,3)(2,8,6,7)(9,13,15,19)(10,18,16,14)(11,21,12,23)(17,22,20,24)(25,26)(27,28)(29,33)(30,34)(31,35)(32,36)(37,39)(38,40);(9,14,15,18)(10,19,16,13)(11,22,12,24)(17,21,20,23)(25,30,31,33)(26,34,35,29)(27,38,32,39)(28,40,36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=g,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

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, 7619609);