#7653886

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	#35063	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4656 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)(11,17)(15,18)(19,23)(22,24)(25,26,30,31)(27,32,34,29)(28,33,37,39)(35,36,40,38);(1,2)(3,6)(4,5)(7,8)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(26,31)(27,34)(33,39)(35,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27)(26,32)(28,35)(29,31)(30,34)(33,36)(37,40)(38,39);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23);(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,36)(34,40);(1,3,4,7)(2,6,5,8)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,32)(26,27,31,34)(28,36,37,38)(33,40,39,35)

Polycyclic presentation:

2^9|a^2=i,b^2=h,e^2=h,f^2=g,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, 7561082);