#8688223

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

4068 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,d^2=h,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, 8595419);