#8093194

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

4570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 8000390);