#135604

order	512
nilpotency class	4
isoclinism	isoclinic class of #135598
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1823
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

329 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,10,12,6)(4,9)(5,8)(7,11,14,16)(13,15)(17,18,21,26,22,28,36,27)(19,23,30,39,31,41,47,40)(20,32,34,25,35,24,33,42)(29,43,45,38,46,37,44,48);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,16)(14,15)(17,19,22,31)(18,23,28,41)(20,29,35,46)(21,30,36,47)(24,37,32,43)(25,38,42,48)(26,39,27,40)(33,44,34,45);(1,3)(2,7,8,14)(4,11)(5,12)(6,9,10,15)(13,16)(18,24)(20,33)(21,36)(23,37)(25,26)(27,42)(28,32)(29,44)(30,47)(34,35)(38,39)(40,48)(41,43)(45,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=d,e^2=gh,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=ehi,f^c=fgi,h^c=hi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42800);