#8424813

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4439 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332009);