#8093161

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

4650 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:

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

Polycyclic presentation:

2^9|c^2=gh,e^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, 8000357);