#7917285

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4440 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)(9,10,15,17)(11,21,22,16)(12,14,20,18)(13,24,23,19)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,23)(13,20)(14,24)(15,22)(17,21)(18,19);(10,17)(11,22)(12,20)(19,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,3,4,7)(2,6,5,8)(9,12,15,20)(10,18,17,14)(11,23,22,13)(16,19,21,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(9,13,15,23)(10,19,17,24)(11,20,22,12)(14,16,18,21)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(9,14,15,18)(10,20,17,12)(11,24,22,19)(13,21,23,16)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7824481);