#142972

order	512
nilpotency class	4
isoclinism	isoclinic class of #142972
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

521 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

2^9|d^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fg,h^b=hi,d^c=dh,e^c=ei,f^c=fg,f^d=fi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50168);