#8107401

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	#35077	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4598 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=ghi,e^2=h,f^2=h,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, 8014597);