#8093689

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

4625 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)(10,14)(11,17)(16,21)(18,23)(25,26,31,32)(27,30,28,33)(29,34,38,39)(35,37,36,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(26,32)(27,28)(34,39)(35,36);(2,5)(6,8)(12,19)(16,21)(18,23)(22,24)(25,27,31,28)(26,33,32,30)(29,36,38,35)(34,37,39,40);(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,31,27)(26,30,32,33)(29,36,38,35)(34,37,39,40);(1,2)(3,6)(4,5)(7,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,31,38)(26,34,32,39)(27,35,28,36)(30,37,33,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,30)(26,27)(28,32)(29,37)(31,33)(34,35)(36,39)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000885);