#8977375

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
prime radical	#42085	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4239 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8884571);