#484570

order	512
nilpotency class	4
isoclinism	isoclinic class of #484570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#60771	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

835 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 835 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,e^2=g,g^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gi,h^a=hi,e^b=egi,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391766);