#146270

order	512
nilpotency class	4
isoclinism	isoclinic class of #146219
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#495
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

225 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=dgh,b^2=deg,c^2=g,d^2=gi,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,d^c=dgi,e^c=egi,f^c=fh,h^c=hi

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, 53466);