#7709399

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

4320 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=g,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, 7616595);