#128833

order	512
nilpotency class	4
isoclinism	isoclinic class of #11769
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
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×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

446 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 104 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)(6,12,14,28)(8,21)(9,10)(13,27,30,29)(15,31,26,22)(16,32,33,36)(17,34)(19,20)(23,35)(24,25);(5,6)(7,16,18,33)(8,22,24,31)(9,26)(10,15)(11,14)(12,25,28,21)(13,20,30,19)(17,27)(23,29)(32,34)(35,36);(1,3,4,2)(5,7,11,18)(6,13)(8,23,24,17)(9,19,10,20)(12,27)(14,30)(15,16)(21,35,25,34)(22,36)(26,33)(28,29)(31,32)

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=ch,d^b=dh,e^b=ei,g^b=gi,d^c=di,f^c=fi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36029);