#9058850

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4262 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=gh,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, 8966046);