#465973

order	512
nilpotency class	3
isoclinism	isoclinic class of #456505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

883 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fg,d^2=fi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,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, 373169);