#98130

order	512
nilpotency class	3
isoclinism	isoclinic class of #97300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

485 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 485 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=eg,f^c=fh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5326);