#8926678

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

4442 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,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, 8833874);