#8093729

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

4474 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)(11,16)(12,13)(14,15)(17,21)(18,22)(19,20)(23,24)(25,26,31,33)(27,34,28,32)(29,39,40,30)(35,36,37,38);(1,2)(3,6)(4,5)(7,8)(11,18)(16,22)(19,23)(20,24)(25,27,31,28)(26,32,33,34)(29,35,40,37)(30,36,39,38);(2,5)(6,8)(10,15)(12,17)(16,22)(19,23)(25,28,31,27)(26,32,33,34)(29,37,40,35)(30,36,39,38);(9,11)(10,16)(12,19)(13,20)(14,18)(15,22)(17,23)(21,24)(25,29)(26,30)(27,35)(28,37)(31,40)(32,36)(33,39)(34,38);(1,2)(3,6)(4,5)(7,8)(9,12,14,17)(10,13,15,21)(11,19,18,23)(16,20,22,24)(25,30)(26,29)(27,36)(28,38)(31,39)(32,35)(33,40)(34,37);(1,3)(2,6)(4,7)(5,8)(9,13)(10,17)(11,20)(12,15)(14,21)(16,23)(18,24)(19,22)(25,29)(26,30)(27,35)(28,37)(31,40)(32,36)(33,39)(34,38)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=h,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, 8000925);