#8107495

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

4504 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014691);