#139370

order	512
nilpotency class	4
isoclinism	isoclinic class of #139350
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

366 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=ghi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fh,d^c=dh,f^c=fh

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, 46566);