#7660519

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	#52249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4354 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)(25,26)(27,32)(28,36)(29,33)(30,35)(31,37)(34,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(10,14)(15,21)(16,22)(23,24)(25,28,29,37)(26,31,33,36)(27,34,35,40)(30,38,32,39);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,28)(26,31)(27,34)(29,37)(30,38)(32,39)(33,36)(35,40);(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,38)(37,40);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,26)(27,30)(28,31)(29,33)(32,35)(34,38)(36,37)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7567715);