#146386

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1797
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1116 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53582);