#9040451

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	#47946	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4589 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=gh,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,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, 8947647);