#419879

order	512
nilpotency class	3
isoclinism	isoclinic class of #418458
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

1543 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)(7,18)(8,14)(9,19)(10,32)(11,16)(12,17)(13,21)(15,31)(20,33)(22,34)(23,25)(24,30)(26,35)(27,28)(29,36);(5,7)(6,13)(8,24)(9,28)(10,22)(11,21)(12,23)(14,31)(15,33)(16,25)(17,34)(18,32)(19,26)(20,29)(27,36)(30,35);(2,4)(5,8,12,27)(6,14,17,28)(7,19,23,15)(9,25,31,18)(10,26,11,33)(13,30,34,29)(16,20,32,35)(21,24,22,36);(5,9,10,29,12,31,11,30)(6,15,16,36,17,19,32,24)(7,20,21,28,23,35,22,14)(8,25,26,13,27,18,33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327075);