#354356

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1176 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=g,e^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=di,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261552);