#7641910

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4422 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=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, 7549106);