#142453

order	512
nilpotency class	4
isoclinism	isoclinic class of #142228
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	C₂⁴	#121
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

504 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49649);