#7709388

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

4570 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,e^2=h,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, 7616584);