#365579

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272775);