#139905

order	512
nilpotency class	4
isoclinism	isoclinic class of #139783
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

426 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=hi,d^2=i,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,c^b=ci,d^b=dgi,e^b=ehi,f^b=fgi,d^c=dh,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47101);