#138453

order	512
nilpotency class	4
isoclinism	isoclinic class of #138415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#19255	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

382 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,d^2=ghi,e^2=gi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egi,f^b=fh,h^b=hi,d^c=dgi,e^c=egi,f^c=fgi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45649);