#135094

order	512
nilpotency class	4
isoclinism	isoclinic class of #135093
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1802
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#459
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

288 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42290);