#467420

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=g,c^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 374616);