#8097731

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

4525 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)(9,10,16,20)(11,13,17,12)(14,18,21,23)(15,19,24,22)(26,32)(28,33)(34,39)(36,40);(1,3,5,8)(2,6,7,4)(9,11,16,17)(10,12,20,13)(14,15,21,24)(18,22,23,19)(25,26,31,32)(27,33,29,28)(30,34,38,39)(35,40,37,36);(1,3)(2,4)(5,8)(6,7)(9,12,16,13)(10,17,20,11)(14,22,21,19)(15,18,24,23)(25,27,31,29)(26,33,32,28)(30,35,38,37)(34,40,39,36);(1,4)(2,3)(5,6)(7,8)(9,13,16,12)(10,11,20,17)(14,19,21,22)(15,23,24,18)(25,28)(26,27)(29,32)(30,36)(31,33)(34,35)(37,39)(38,40);(9,14)(10,18)(11,15)(12,22)(13,19)(16,21)(17,24)(20,23)(25,29,31,27)(26,28,32,33)(30,35,38,37)(34,40,39,36);(9,15)(10,19)(11,21)(12,18)(13,23)(14,17)(16,24)(20,22)(25,30,31,38)(26,34,32,39)(27,35,29,37)(28,36,33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,e^2=gi,f^2=gi,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, 8004927);