#7642625

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	#35064	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4554 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)(11,12)(13,21)(14,16)(15,19)(17,22)(18,23)(20,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,23)(20,21)(22,24)(26,30)(31,33)(32,36)(38,39);(10,16)(12,15)(17,21)(20,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(9,11)(10,15)(12,16)(13,18)(14,19)(17,23)(20,21)(22,24);(1,3,4,7)(2,6,5,8)(9,12,14,15)(10,11,16,19)(13,20,22,23)(17,18,21,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(9,13,14,22)(10,17,16,21)(11,18,19,24)(12,20,15,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|b^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

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, 7549821);