#9018026

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	#41955	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4459 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)(3,7)(4,5)(6,8)(25,26)(27,29)(28,37)(30,38)(31,33)(32,36)(34,35)(39,40);(3,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27,31,36)(26,32,33,29)(28,34,38,40)(30,35,37,39);(1,3,5,8)(2,6,4,7)(10,14)(11,17)(12,20)(22,24)(25,28,31,38)(26,30,33,37)(27,34,36,40)(29,39,32,35);(3,8)(6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28)(26,30)(27,34)(29,39)(31,38)(32,35)(33,37)(36,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,29,31,32)(26,27,33,36)(28,35,38,39)(30,40,37,34);(1,4,5,2)(3,6,8,7)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,30)(26,28)(27,35)(29,40)(31,37)(32,34)(33,38)(36,39)

Polycyclic presentation:

2^9|b^2=g,c^2=h,e^2=g,f^2=i,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, 8925222);