#126068

order	512
nilpotency class	4
isoclinism	isoclinic class of #125709
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

255 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 54 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)(4,8,5,13)(7,14,16,11)(17,18)(19,28)(20,24)(21,36)(22,26)(23,29)(25,33)(27,35)(30,42)(31,39)(32,37)(34,40)(38,47)(41,48)(43,46)(44,45);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,22,29)(18,23)(20,32,34,30)(21,31,35,45)(24,37)(25,38)(26,28)(27,39)(33,47,48,46)(36,44)(40,42)(41,43);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,24)(20,33)(21,35)(23,37)(25,27)(26,40)(28,42)(30,46)(31,45)(32,47)(34,48)(36,41)(38,39)(43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=f,d^2=f,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,h^a=hi,d^b=dfi,e^b=egi,d^c=dg,e^c=eh,h^c=hi,e^d=ei

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33264);