#144560

order	512
nilpotency class	4
isoclinism	isoclinic class of #144523
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	C₂×C₄×C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

271 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=dh,c^2=h,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,e^b=eh,f^b=fhi,d^c=di,e^c=egi,f^c=fh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51756);