#139684

order	512
nilpotency class	4
isoclinism	isoclinic class of #139558
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×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

364 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,c^b=ci,e^b=eh,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, 46880);