#9058825

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	#43033	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4439 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,e^2=h,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, 8966021);