#463939

order	512
nilpotency class	3
isoclinism	isoclinic class of #460861
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

814 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)(5,6,10,14,11,16,9,15)(7,17,19,29,21,28,20,12)(8,22,25,13,26,31,24,32)(18,33,34,36,35,27,23,30);(5,7,10,20,11,21,9,19)(6,12,15,17,16,29,14,28)(8,23,25,35,26,34,24,18)(13,30,22,33,32,36,31,27);(5,8,10,25,11,26,9,24)(6,13,15,22,16,32,14,31)(7,18,20,23,21,35,19,34)(12,27,17,30,29,33,28,36);(2,4)(6,14,16,15)(8,24,26,25)(12,28,29,17)(18,34,35,23)

Polycyclic presentation:

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

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, 371135);