#483843

order	512
nilpotency class	4
isoclinism	isoclinic class of #483839
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	C₂²×D₁₆	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

478 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,d^a=di,e^a=eg,g^a=gi,c^b=ch,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	#72216715, #72216720, #72216726, …
maximal normal subgroup of	#72216715, #72216720, #72216726, …
maximal quotient of	C₂×#483843, #72318456, #72321373

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391039);