#480234

order	512
nilpotency class	3
isoclinism	isoclinic class of #2883
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₁₆
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂²×#41	C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

867 conjugacy classes of subgroups; only the 356 normal subgroups are shown.

normal subgroups only

This group has 867 conjugacy classes of subgroups — too many to draw. Only its 356 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) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=g,f^2=h,h^2=i,b^a=be,e^a=eg,f^b=fg,d^c=dg

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387430);