#8184785

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

4516 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8091981);