#7654756

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	#34909	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4472 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=h,e^2=gi,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, 7561952);