#144572

order	512
nilpotency class	4
isoclinism	isoclinic class of #144561
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2023
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

296 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=dg,c^2=i,e^2=ghi,f^2=g,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fg,h^a=hi,c^b=cf,d^c=dg,e^c=egh,f^c=fg,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51768);