#144605

order	512
nilpotency class	4
isoclinism	isoclinic class of #144603
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2023
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

316 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 62 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)(3,9,11,15)(4,7)(5,8)(6,12,10,13)(14,16)(17,18,20,25)(19,28,30,23)(21,24,33,42)(22,37,34,27)(26,44,43,36)(29,40,32,39)(31,41,35,48)(38,46,45,47);(1,2)(3,10,11,6)(4,7)(5,8)(9,12,15,13)(14,16)(17,19)(18,23)(20,30)(21,35)(22,29)(24,41)(25,28)(26,45)(27,40)(31,33)(32,34)(36,47)(37,39)(38,43)(42,48)(44,46);(1,3)(2,6)(4,13)(5,11)(7,15)(8,10)(9,16)(12,14)(18,24)(19,29)(21,36)(22,34)(23,38)(25,42)(26,27)(28,45)(30,32)(31,35)(33,44)(37,43)(39,48)(40,41)

Polycyclic presentation:

2^9|a^2=d,b^2=di,e^2=ghi,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fgi,h^a=hi,c^b=cf,e^b=ei,f^b=fgi,d^c=dg,e^c=egh,f^c=fi,h^c=hi

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, 51801);