#128351

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	C₂×#324
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

416 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

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, 35547);