#477877

order	512
nilpotency class	3
isoclinism	isoclinic class of #476106
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

862 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)(3,8,4,6)(9,10,14,19)(11,20,23,35)(12,17,26,34)(13,16,28,18)(15,27,31,22)(21,38,24,37)(25,36,29,32)(30,40,33,39);(1,3,5,4)(2,6,7,8)(9,11,14,23)(10,15,19,31)(12,24,26,21)(13,27,28,22)(16,20,18,35)(17,33,34,30)(25,39,29,40)(32,38,36,37);(1,4,5,3)(2,6,7,8)(10,16)(12,25)(13,28)(15,20)(17,34)(18,19)(21,39)(22,27)(24,40)(26,29)(31,35)(37,38);(3,4)(6,8)(9,12)(10,17)(11,21)(13,29)(14,26)(15,30)(16,32)(18,36)(19,34)(20,37)(22,40)(23,24)(25,28)(27,39)(31,33)(35,38)

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=efi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385073);