#123417

order	512
nilpotency class	4
isoclinism	isoclinic class of #123412
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₁₆

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1712
commutator, Frattini, cosocle	C₈²	C₂³
prime radical	#1712	C₂²
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

225 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=dfi,c^2=egh,d^2=fh,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gi,c^b=ch

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225196, #72225197, #72225200
maximal normal subgroup of	#72225196, #72225197, #72225200

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30613);