#7709810

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

4347 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=g,e^2=gh,f^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, 7617006);