#465976

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

917 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)(3,6)(4,8)(5,7)(9,10)(11,17)(12,18)(13,20)(14,19)(15,21)(16,23)(22,37)(24,32)(25,31)(26,30)(27,36)(28,35)(29,34)(33,38)(39,40);(1,3,5,4)(2,6,7,8)(9,11,14,24,15,25,13,23)(10,16,20,31,21,32,19,17)(12,22,27,26,28,38,29,39)(18,30,34,37,35,40,36,33);(2,7)(6,8)(9,11)(10,17)(12,26)(13,24)(14,23)(15,25)(16,19)(18,30)(20,32)(21,31)(22,27)(28,39)(29,38)(33,34)(35,40)(36,37);(1,4,5,3)(2,8,7,6)(9,12)(10,18)(11,22)(13,29)(14,27)(15,28)(16,30)(17,33)(19,36)(20,34)(21,35)(23,39)(24,26)(25,38)(31,37)(32,40)

Polycyclic presentation:

2^9|b^2=fg,d^2=f,e^2=i,g^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,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, 373172);