#142992

order	512
nilpotency class	4
isoclinism	isoclinic class of #142975
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#18356	C₂
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 469 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=g,e^2=i,f^2=gi,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=fgi,h^b=hi,d^c=dh,e^c=ei,f^c=fgi,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, 50188);