#480052

order	512
nilpotency class	4
isoclinism	isoclinic class of #33359
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#99
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂²×#324
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ C₂×#568 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

969 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 969 conjugacy classes of subgroups — too many to draw. Only its 312 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=h,h^2=i,b^a=be,e^a=eg,g^a=gi,f^b=fgi,d^c=di,f^e=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387248);