#139636

order	512
nilpotency class	4
isoclinism	isoclinic class of #139532
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	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#534	C₂×C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 502 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,4,7,5,8,3,6)(9,10,14,19)(11,20,24,37)(12,25,21,17)(13,27)(15,31,34,22)(16,29)(18,30)(23,32)(26,35)(28,36)(33,39)(38,40);(2,6,8,7)(9,11,21,31,14,24,12,22)(10,15,25,37,19,34,17,20)(13,28,29,38,30,23,26,39)(16,32,27,40,35,36,18,33);(1,3,5,4)(2,6,8,7)(9,12,14,21)(10,16,19,35)(11,22,24,31)(13,26,30,29)(15,32,34,36)(17,18,25,27)(20,33,37,40)(23,38,28,39)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ehi,d^c=dh,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46832);