#484652

order	512
nilpotency class	4
isoclinism	isoclinic class of #484652
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1598
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

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

normal subgroups only

This group has 903 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)(3,12)(4,7)(5,20)(6,11)(8,13)(9,16)(10,18)(14,26)(15,25)(17,21)(19,22)(23,32)(24,31)(27,30)(28,29);(2,7)(4,16)(5,18)(8,21)(9,10)(11,22)(12,26)(14,30)(15,29)(17,31)(19,32)(20,23)(24,25)(27,28);(1,3,6,13)(2,8)(4,17,19,14)(5,15,18,29)(7,21)(9,24)(10,25)(11,12)(16,31,32,30)(20,28)(22,26)(23,27);(3,13)(8,12)(14,17)(15,29)(21,26)(24,27)(25,28)(30,31)

Polycyclic presentation:

2^9|c^2=h,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^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, 391848);