#7710087

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×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4422 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)(12,19)(16,21)(17,23)(22,24)(25,26)(27,29)(28,34)(30,37)(31,32)(33,35)(36,39)(38,40);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(26,32)(27,35)(34,40)(36,37);(1,3,4,7)(2,6,5,8)(10,14)(15,20)(16,21)(22,24)(25,27)(26,33)(28,36)(29,32)(30,34)(31,35)(37,38)(39,40);(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,28,31,38)(26,34,32,40)(27,37,35,36)(29,30,33,39);(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,29,31,33)(26,35,32,27)(28,30,38,39)(34,36,40,37);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,30)(26,36)(27,34)(28,33)(29,38)(31,39)(32,37)(35,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=i,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, 7617283);