#142260

order	512
nilpotency class	4
isoclinism	isoclinic class of #142260
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9914
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

739 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 739 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,g^b=gi,h^b=hi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49456);