#8926673

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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=g,e^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, 8833869);