#143395

order	512
nilpotency class	4
isoclinism	isoclinic class of #143211
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
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 ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

362 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

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

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, 50591);