#7734370

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×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4517 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)(3,5,7,8)(9,10)(11,19)(12,22)(13,18)(14,16)(15,21)(17,23)(20,24)(25,26,30,33)(27,34,31,28)(29,32,38,40)(35,39,37,36);(1,2)(3,5)(4,6)(7,8)(9,11,14,21)(10,15,16,19)(12,20,23,13)(17,24,22,18)(25,27,30,31)(26,28,33,34)(29,37,38,35)(32,39,40,36);(10,16)(15,19)(17,22)(18,24)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,38)(39,40);(9,10)(11,15)(12,17)(13,18)(14,16)(19,21)(20,24)(22,23);(9,12)(10,17)(11,20)(13,21)(14,23)(15,24)(16,22)(18,19)(25,28)(26,31)(27,33)(29,36)(30,34)(32,37)(35,40)(38,39);(1,3,4,7)(2,5,6,8)(9,13,14,20)(10,18,16,24)(11,12,21,23)(15,17,19,22)(25,29,30,38)(26,32,33,40)(27,35,31,37)(28,36,34,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7641566);