#128805

order	512
nilpotency class	4
isoclinism	isoclinic class of #11836
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1514
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

408 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ei,d^2=g,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,d^c=di,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36001);