#7709537

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

4384 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=h,e^2=gh,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, 7616733);