#145520

order	512
nilpotency class	4
isoclinism	isoclinic class of #145502
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33811	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 278 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)(3,10)(4,12)(5,13)(6,9)(7,11)(8,15)(14,16)(17,18)(19,33)(20,26)(21,41)(22,28)(23,31)(24,30)(25,32)(27,36)(29,40)(34,46)(35,45)(37,43)(38,47)(39,44)(42,48);(1,3)(2,7)(4,8)(5,14)(6,12)(9,16)(10,13)(11,15)(18,26)(19,34)(20,22)(21,40)(23,24)(27,42)(28,30)(29,32)(31,41)(33,44)(35,38)(36,39)(37,48)(45,46);(1,4,5,7,6,10,15,16)(2,3,8,11,9,12,13,14)(17,19,21,34,23,37,20,35,25,39,22,36,24,38,40,48)(18,27,29,44,30,45,28,43,32,46,26,33,31,42,41,47)

Polycyclic presentation:

2^9|c^2=dh,d^2=gi,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egi,f^b=fh,g^b=gi,h^b=hi,e^c=eghi,f^c=fghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52716);