#146910

order	512
nilpotency class	4
isoclinism	isoclinic class of #12077
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1762
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 54106);