#419834

order	512
nilpotency class	3
isoclinism	isoclinic class of #419461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1589 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,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, 327030);