#7655517

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4500 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)(11,18)(12,20)(13,15)(14,17)(16,22)(19,23)(21,24)(25,26)(27,34)(28,29)(30,33)(31,35)(32,38)(36,37)(39,40);(1,2)(3,6)(4,5)(7,8)(25,27)(26,32)(28,39)(29,37)(30,36)(31,38)(33,40)(34,35);(9,11,14,13)(10,15,17,18)(12,19,22,21)(16,23,20,24)(25,28,31,30)(26,29,35,33)(27,36,38,39)(32,40,34,37);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,29,31,33)(26,28,35,30)(27,37,38,40)(32,39,34,36);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,15,17,18)(12,21,22,19)(16,23,20,24)(25,30,31,28)(26,33,35,29)(27,36,38,39)(32,40,34,37);(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,22)(21,24)(25,27)(26,34)(28,36)(29,37)(30,39)(31,38)(32,35)(33,40)

Polycyclic presentation:

2^9|c^2=hi,d^2=i,e^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

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, 7562713);