#405575

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	#23443	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1432 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312771);