#142993

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	#534	D₄
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 438 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) = 128

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,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, 50189);