#484233

order	512
nilpotency class	4
isoclinism	isoclinic class of #484233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1595
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1103 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 1103 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)(3,14)(4,8)(5,12)(6,11)(7,13)(9,17)(10,18)(15,28)(16,27)(19,24)(20,25)(21,23)(22,32)(26,30)(29,31);(2,8)(4,18)(5,6)(9,22)(10,12)(11,23)(13,24)(14,27)(16,31)(17,20)(19,21)(25,29)(26,28)(30,32);(1,3)(2,9)(4,16)(5,20)(6,17)(7,15)(8,22)(10,27)(11,28)(12,14)(13,25)(18,31)(19,30)(21,32)(23,26)(24,29);(3,15)(9,25)(14,28)(16,30)(17,20)(22,29)(26,27)(31,32)

Polycyclic presentation:

2^9|e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,g^a=gi,h^a=hi,e^b=eg,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^c=fi,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, 391429);