#7654713

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	#40007	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4403 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)(9,10,14,18)(11,13,17,15)(12,20,22,16)(19,23,24,21)(25,26,32,30)(27,36,39,33)(28,29,34,38)(31,35,37,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,21)(22,24)(25,27)(26,33)(28,40)(29,37)(30,36)(31,38)(32,39)(34,35);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,21)(22,24)(25,28)(26,29)(27,35)(30,38)(31,33)(32,34)(36,37)(39,40);(1,4,5,3)(2,7,8,6)(9,12,14,22)(10,16,18,20)(11,19,17,24)(13,21,15,23)(25,29,32,38)(26,34,30,28)(27,37,39,31)(33,35,36,40);(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,23)(22,24)(25,30,32,26)(27,36,39,33)(28,38,34,29)(31,35,37,40);(9,13)(10,17)(11,18)(12,21)(14,15)(16,24)(19,20)(22,23)(25,31,32,37)(26,35,30,40)(27,38,39,29)(28,33,34,36)

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=g,e^2=i,f^2=i,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, 7561909);