#483889

order	512
nilpotency class	4
isoclinism	isoclinic class of #483882
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

429 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 160 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 — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dhi,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321142, #72321157, #72321354, …
maximal normal subgroup of	#72321142, #72321157, #72321354, …
maximal quotient of	#72344014

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391085);