#458085

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

921 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:

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

Polycyclic presentation:

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

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, 365281);