#8924802

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

4779 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)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(27,33)(31,38)(34,36)(39,40);(3,7)(6,8)(10,14)(15,17)(16,21)(22,23)(25,26)(27,31)(28,35)(29,30)(32,37)(33,38)(34,40)(36,39);(1,2)(3,6)(4,5)(7,8)(12,20)(16,21)(18,24)(22,23)(26,30)(31,38)(32,35)(39,40);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,27)(26,31)(28,36)(29,33)(30,38)(32,40)(34,37)(35,39);(1,3)(2,6)(4,7)(5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28)(26,32)(27,34)(29,37)(30,35)(31,39)(33,36)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,37)(36,40)

Polycyclic presentation:

2^9|d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8831998);