#8688203

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4031 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595399);