#8096570

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

4226 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:

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

Polycyclic presentation:

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

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, 8003766);