#9058258

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	#44224	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4382 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965454);