#459433

order	512
nilpotency class	3
isoclinism	isoclinic class of #454372
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
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

1104 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,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, 366629);