#459768

order	512
nilpotency class	3
isoclinism	isoclinic class of #454041
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

943 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:

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

Polycyclic presentation:

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

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, 366964);