#8751639

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

4059 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8658835);