#8096540

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

4235 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^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, 8003736);