#135244

order	512
nilpotency class	4
isoclinism	isoclinic class of #129430
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	#1820
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

299 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,28,31,26)(20,32)(21,25)(22,27)(23,30,37,44)(24,29,39,43)(33,36)(34,41)(35,40)(38,42,47,48)(45,46);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,19)(18,23,27,37)(20,29)(21,30)(22,31)(24,36,39,46)(25,28,40,26)(32,38,41,47)(33,42)(34,43)(35,44)(45,48);(2,7)(3,9)(4,11)(13,16)(17,20,21,33,22,34,35,45)(18,24,25,38,27,39,40,47)(19,29,30,42,31,43,44,48)(23,36,28,41,37,46,26,32)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=d,d^2=i,f^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ch,f^b=fi,f^c=fg,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42440);