#7711124

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	#59217	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4396 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

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, 7618320);