#470278

order	512
nilpotency class	3
isoclinism	isoclinic class of #459488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
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

1173 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=eg,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,d^c=de,e^c=ei,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, 377474);