#466297

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×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

882 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)(9,10,14,20)(11,22,25,17)(12,27,30,18)(13,21,15,19)(16,26,33,24)(23,38,39,34)(28,35,40,32)(29,36,31,37);(1,3,5,4)(2,6,7,8)(9,11,13,24,14,25,15,26)(10,16,19,22,20,33,21,17)(12,23,31,40,30,39,29,28)(18,32,37,38,27,35,36,34);(2,7)(6,8)(9,11,14,25)(10,17,20,22)(12,28,30,40)(13,26,15,24)(16,19,33,21)(18,35,27,32)(23,31,39,29)(34,37,38,36);(1,4,5,3)(2,8,7,6)(9,12,14,30)(10,18,20,27)(11,23,25,39)(13,31,15,29)(16,32,33,35)(17,34,22,38)(19,37,21,36)(24,40,26,28)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=i,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, 373493);