#484227

order	512
nilpotency class	4
isoclinism	isoclinic class of #484227
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

1119 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 1119 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)(3,8)(4,7)(5,19)(6,11)(9,15)(10,17)(12,23)(13,20)(14,28)(16,31)(18,21)(22,30)(24,27)(25,26)(29,32);(2,7)(4,15)(5,17)(8,20)(9,10)(11,21)(13,27)(14,26)(16,29)(18,30)(19,22)(23,31)(24,25)(28,32);(1,3)(2,8)(4,16)(5,14)(6,12)(7,20)(9,24)(10,25)(11,23)(13,18)(15,29)(17,26)(19,28)(21,31)(22,32)(27,30);(3,12)(8,23)(13,16)(14,26)(20,31)(24,32)(25,28)(27,29)

Polycyclic presentation:

2^9|e^2=gi,g^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gi,h^a=hi,e^b=eg,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, 391423);