#8107576

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	C₂×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4450 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=gi,e^2=h,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, 8014772);