#147333

order	512
nilpotency class	4
isoclinism	isoclinic class of #2196
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator	C₂×D₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	#1867
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

699 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=i,b^a=bd,c^a=ce,h^a=hi,e^b=ehi,d^c=dh,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54529);