#465311

order	512
nilpotency class	3
isoclinism	isoclinic class of #465214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

825 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)(6,11)(7,15)(8,19)(9,22)(12,24)(13,29)(14,26)(16,28)(17,33)(20,31)(21,34)(30,35);(5,6,9,11,10,14,22,26)(7,16,23,28,15,12,18,24)(8,13,19,25,21,29,34,36)(17,32,33,30,31,27,20,35);(5,7,9,23,10,15,22,18)(6,12,26,28,14,16,11,24)(8,20,19,17,21,33,34,31)(13,30,36,32,29,35,25,27);(2,4)(5,8,9,19,10,21,22,34)(6,13,11,25,14,29,26,36)(7,17,18,20,15,31,23,33)(12,27,28,30,16,32,24,35)

Polycyclic presentation:

2^9|b^2=eg,c^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=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, 372507);