#419845

order	512
nilpotency class	3
isoclinism	isoclinic class of #419843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1564 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)(5,6,9,16,11,15,10,14)(7,17,19,28,21,12,20,29)(8,22,24,31,25,32,26,13)(18,30,23,27,34,36,35,33);(5,7,11,21)(6,12,15,17)(8,23,25,35)(9,20,10,19)(13,30,31,36)(14,29,16,28)(18,26,34,24)(22,33,32,27);(2,4)(5,8)(6,13)(7,18)(9,26)(10,24)(11,25)(12,27)(14,22)(15,31)(16,32)(17,33)(19,35)(20,23)(21,34)(28,36)(29,30);(6,14,15,16)(12,28,17,29)(13,22,31,32)(27,36,33,30)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=e,f^2=i,g^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^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi

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, 327041);